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Randomized Δ-edge colouring viaexchanges of complex coloursTony T. Lee a b , Yujie Wan a & Hao Guan aa Department of Information Engineering , The Chinese Universityof Hong Kong , Hong Kongb Department of Electronic Engineering , Shanghai Jiao TongUniversity , Shanghai , ChinaAccepted author version posted online: 03 Sep 2012.Publishedonline: 08 Oct 2012.

To cite this article: Tony T. Lee , Yujie Wan & Hao Guan (2013) Randomized Δ-edge colouring viaexchanges of complex colours, International Journal of Computer Mathematics, 90:2, 228-245, DOI:10.1080/00207160.2012.726712

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International Journal of Computer Mathematics, 2013Vol. 90, No. 2, 228–245, http://dx.doi.org/10.1080/00207160.2012.726712

Randomized �-edge colouring via exchanges of complex colours

Tony T. Leea,b*, Yujie Wana and Hao Guana

aDepartment of Information Engineering, The Chinese University of Hong Kong, Hong Kong;bDepartment of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China

(Received 25 April 2012; revised version received 21 August 2012; accepted 29 August 2012 )

This paper explores the application of a new algebraic method of colour exchanges to the edge colouringof simple graphs. Vizing’s theorem states that the edge colouring of a simple graph G requires either � or� + 1 colours, where � is the maximum vertex degree of G. Holyer proved that it is NP-complete to decidewhether G is �-edge colourable even for cubic graphs. By introducing the concept of complex colourededges, we show that the colour-exchange operation of complex colours follows the same multiplicationrules as quaternion. An initially �-edge-coloured graph G allows variable-coloured edges, which can beeliminated by colour exchanges in a manner similar to variable eliminations in solving systems of linearequations. The problem is solved if all variables are eliminated and a properly �-edge-coloured graphis reached. For �-regular uniform random graphs, we prove that our algorithm returns a proper �-edgecolouring with a probability of 1 − 1/n in O(n3

0�4n4) time if G is �-edge colourable. Otherwise, the

algorithm halts in polynomial time and signals the impossibility of a solution, meaning that the chromaticindex of G probably equals � + 1.

Keywords: incidence graph; edge colouring; colour exchange; Kempe path

2010 AMS Subject Classification: 05C15

1. Introduction

The chromatic index χe(G) of a simple graph G = (V , E), with vertex set V and edge set E, isthe minimum number of colours required to colour the edges of the graph such that no adjacentedges have the same colour. A theorem proved by Vizing [21] states that the chromatic index iseither � or � + 1, where � is the maximum vertex degree of graph G. The graph G is said to beClass 1 if χe(G) = �; otherwise, it is Class 2. Except for some particular types of graphs, such asbipartite graphs, it is inherently difficult to classify an arbitrary simple graph. In fact, Holyer hasproved in [15] that it is NP-complete to determine the chromatic index of arbitrary simple grapheven if � = 3.

In this paper, we propose a new algebraic method of colour exchanges for edge colouringof simple graphs. The original colour-exchange method was devised by Alfred Kempe in hisendeavour to prove the four-colour theorem [16]. Although his attempt was unsuccessful, hismethod remains critical to the final proof given by Appel and Haken [4]. The Kempe chain methodwas defined on two-coloured vertices. An extension of this method to two-coloured edges, calledalternating paths, constitutes the basis of the proof of Vizing’s theorem and Edmonds’ matching

*Corresponding author. Email: ttlee@ie.cuhk.edu.hk

© 2013 Taylor & Francis

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International Journal of Computer Mathematics 229

algorithm [9]. By introducing the concept of complex colours, we show that the colour-exchangeoperation performed on alternating paths follows the same multiplication rules as quaternion, andedge colouring is a procedure of variable eliminations.

We consider each edge e ∈ E as a pair of links; each link is a half-edge. Let C be the set ofcolours. The colouring of graph G = (V , E) is a function c : E �→ C × C defined by assigning acolour pair, or a complex colour, to each e ∈ E, one colour assigned for each link. If |C| = �,then a colour configuration c of G such that all links incident to the same vertex having differentcolours can be easily obtained. Suppose that the complex colour assigned to edge e is c(e) =(α, β), α, β ∈ C, then the coloured edge e is a variable if α �= β; otherwise, c(e) = (α, α) is aconstant for any colour α.A proper �-edge colouring of graph G can be achieved by eliminating allvariables.

Our edge-colouring algorithm starts with an arbitrary colour configuration c of G, which maycontain variable edges. Applying a sequence of well-defined ‘moves’ (colour exchanges) of vari-ables in a configuration may lead to other new configurations. Variables can be systematicallyeliminated when they encounter other variables while moving around the graph. For a graph Gwith � ≥ 3, the algorithm is initialized by an arbitrary configuration of G with a set of � coloursC = {c1, . . . , c�}. First, we eliminate variables that contain colour c1, then remove the remainingvariables that contain colour c2, and so on. The problem is solved if all variables are eliminatedand a properly �-edge-coloured graph is reached; otherwise, the algorithm halts in polynomialtime and signals the impossibility of a solution, meaning that the chromatic index of G probablyequals � + 1, which implies that the graph G could be class 2.

In principle, each vertex v ∈ V and each edge e ∈ E represent a constraint on colouring ofedges, and the entire graph G can be considered as a set of simultaneous equations. The variableelimination procedure of edge colouring is similar to the Gaussian elimination of solving systemsof linear equations. A comparison between these two procedures is summarized in Table 1.

In spite of the similarity between solving linear equations and edge colouring, the main differ-ence is recognizing the final state. If a system of linear equations has no solution, the inconsistencyof the system can be identified by variable eliminations in polynomial time. However, eliminatingvariable edges of class 2 graphs may result in an infinite loop. Recognizing class 2 cubic graphs,called snarks, could have been significant in determining the halting state of edge-colouring algo-rithms.A polynomial time algorithm for identifying snarks will immediately lead to the conclusionthat P = NP.

Based on random walks (RWs) on graphs, the average time that a variable hits another variablein a graph has a polynomial bound [2,3,17]. For a randomly generated graph G with n vertices, weprove that our algorithm returns a proper �-edge colouring with a probability of at least 1 − 1/nin O(n3

0�4n4) time if G is a class 1 graph with chromatic index χe(G) = �. Otherwise, it returns

with absolute certainty if G is a class 2 graph.In existing literature, the only known exact algorithms are the O(2|V |/2) and O(1.5039|V |)

algorithms for 3-edge colouring proposed by Beigel and Eppstein [5] and Eppstein [10].Vizing’s proof implies an O(|V ||E|) time algorithm with � + 1 colours, which was later

Table 1. Comparison between system of linear equations and edge colouring of simple graphs.

System of linear equations Edge colouring

Operations Arithmetic operations Colour exchangesConstraints Linear equations Vertices and edgesUnknowns Variables Variable-coloured edgesAlgorithms Variable elimination Variable eliminationSolutions Consistency �-colourableNo solution Inconsistency Infinite loop (χe(G) = � + 1)

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improved to O(|E|√|V | log |V |) by Gabow et al. [12]. Some approximation algorithms withhigh probability of success were reported in [1,8,13]. Grable and Panconesi [13] proposed anedge-colouring algorithm using (1 + ε)� colours, which operates in O(log log |V |) rounds if �

is larger than polylog|V | but smaller than any positive power of |V |. Dubhashi et al. [8] pro-posed another O(log |V | + logs |V | log log |V |) time algorithm using � + �/ logs |V | colours if� = �(logk |V |) for some constants s, k > 0. For general multigraph with � = ω(|V |2), Aggar-wal et al. [1] proposed an algorithm using � + o(�) colours that runs in O(|V |2) steps. There arealso some heuristics reported in [14] that do not provide any performance guarantees.

It is well known that bipartite graphs are all class 1 and �-edge colourable. The best-knownalgorithm for finding a proper �-edge colouring of a bipartite graph runs in time O(|E| log �)

[6]. The running time of our algorithm for bipartite graphs is of the order of O(|E| log |V |). Asfor polynomial-time algorithm for finding �-edge colouring of non-bipartite graphs, the onlyknown result is 3-edge colouring of cubic planar graphs. As Tait proved that the 3-edge colouringproblem of bridgeless cubic planar graphs is equivalent to the four colour map problem [20], theimproved proof of four-colour theorem [18] actually can give rise to a quadratic algorithm forfinding proper 3-edge colouring of cubic planar graphs. Thus, our approach is the first randomizedalgorithm for �-edge colouring of general graphs.

The rest of this paper is organized as follows. In Section 2, we establish the rules of colourexchanges and Kempe walks. We also describe the variable elimination method based on Kempewalks and introduce the concept of canonical configurations. Section 3 is devoted to RWs ongraphs. In particular, we give an RW algorithm to eliminate remaining variables in canonicalconfigurations. In addition, this section also contains the performance analysis and experimen-tal results of the RW algorithm. Section 4 consists of a conclusion and discussions on futureresearch. Animations of the edge-colouring algorithms proposed in this paper are posted atYouTube http://www.youtube.com/watch?v=KMnj4UMYl7k.

2. Colour-exchange operations of complex colours

This section introduces an algebraic method of colour exchanges implemented on the edges of acoloured simple graph. We first describe the properties of colour function defined on incidencegraphs and then establish the rules of colour exchanges and Kempe walks.

Let G = (V , E) be a simple graph with vertex set V and edge set E. The incidence graphG∗ is constructed from G by placing a fictitious vertex in the middle of each edge of G. LetE∗(G∗) = {e∗

i,j|ei,j ∈ E(G)} denote the set of fictitious vertices on edges. Then, edge ei,j ∈ E(G∗)consists of two links, denoted by li,j = (vi, e∗

i,j) and lj,i = (vj, e∗i,j) which connect two end vertices

vi and vj of ei,j. Figure 1(a) illustrates the incidence graph of the tetrahedron.Let L(G∗) be the set of links and C = {c1, c2, . . . , c�} denote a set of � colours. A colouring

function c defined on an incidence graph G∗ is a mapping of colour assignments on links c :L(G∗) �→ C. The colour of link li,j ∈ L(G∗) is denoted as c(li,j) = ci,j. Since each edge e ∈ E(G∗)consists of two links, the colour function c can also be considered as a mapping defined on theset of edges: c : E(G∗) �→ C × C.

We define the coloured edge c(ei,j) = ei,j = (ci,j, cj,i) as a two-tuple colour vector, where ci,j =c(li,j) and cj,i = c(lj,i) are respective colours of the two links of ei,j. The following properties of acolour function c are related to the edge colouring of graph G.

Definition 2.1 Let c be a colouring function of the incidence graph G∗.

(i) The coloured edge ei,j = (ci,j, cj,i) is a constant if ci,j = cj,i; otherwise, it is a variable. Thenumber of variables is denoted by nc.

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International Journal of Computer Mathematics 231

(a) (b) (c)

Figure 1. (a) Incidence graph of tetrahedron. (b) Consistent colouring with variables e1,3 = (g, r) and e1,2 = (r, g). (c)Proper colouring of tetrahedron (colour online only).

(ii) Vertex constraint: The colouring function c is consistent if colours assigned to those linksincident to the vertex v are all distinct for all v ∈ V(G∗).

(iii) Edge constraint: The colouring function c is proper if it is consistent and all coloured edgesare constants.

The colouring of incidence graph G∗ is called a configuration of graph G. As an example, aconsistent configuration of a tetrahedron containing two variables is shown in Figure 1(b), and aproper configuration of a tetrahedron is shown in Figure 1(c) with the set of colours C = {r, g, b},where r, g, and b represent red, green, and blue colours, respectively.

The edge constraints are sets of equalities of colours and the vertex constraints are unequalities.The equality (=) is a transitive binary relation but the unequality ( �=) is not. Most edge-colouringalgorithms assign edge colours to satisfy vertex constraints, sets of unequalities. However, math-ematically, it is more natural and usually much easier to solve problems with equalities. Initially,we start with an arbitrary consistent colour function c with � colours that satisfies the vertexconstraint at every vertex of the graph, which may contain variable-coloured edges. Our edge-colouring algorithm is a systematic procedure to eliminate those variables, similar to the procedureof solving a system of simultaneous equations.

2.1 Kempe walks

In general, variable eliminations require a sequence of colour exchanges, called Kempe walks,that are performed on a two-coloured Kempe path defined as follows.

Definition 2.2 In a consistently coloured incidence graph, a (α, β)-Kempe path, or simply(α, β) path, where α, β ∈ C and α �= β, is a sequence of adjacent links l1, l2, . . . , ln such thatc(li) ∈ {α, β} for i = 1, 2, . . . , n. The corresponding sequence of vertices contained in the path isdefined as its interior chain. There are two types of maximum (α, β) path:

(i) (α, β) cycle or closed (α, β) path: The two end-links l1 and ln are adjacent to each other.(ii) Open (α, β) path: Only one α or β coloured link is adjacent to the end-links l1 and ln.

A (α, β)-variable edge is always contained in a maximum (α, β) path, either a (α, β) cycle oran open (α, β) path. Note that the open path may either end at a fictitious vertex or a real vertex.The following lemma can be obtained by simple counting arguments.

Lemma 2.3 An even (odd) (α, β) cycle contains even (odd) number of (α, β) variables.

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Proof Deleting all variables in the (α, β) cycle by edge contraction, the remaining constantedges constitute an even (α, β) cycle. The lemma is established by the following relation:

#variables = #edges − even#constant edges. �

Variable eliminations can be achieved by the following colour-exchange operation performedon adjacent coloured edges.

Definition 2.4 Let ej,i = (cj,i, ci,j) and ei,k = (ci,k , ck,i) be two coloured edges incident to thesame vertex vi, written as (cj,i, ci,j) ◦ (ci,k , ck,i). Suppose that ci,j = β and ci,k = α, for α, β ∈ C ={c1, . . . , c�}; then, the binary operation ⊗ defined below exchanges the colours of links li,j andli,k incident to vi:

(cj,i, ci,j) ⊗ (ci,k , ck,i) = (cj,i, β) ⊗ (α, ck,i) ⇒ (cj,i, α) ◦ (β, ck,i).

The binary operation ⊗ is non-commutative but associative; it can be considered as a trans-formation of a consistent colouring function c to another consistent colouring function c′ suchthat c′

i,j = ci,k = α and c′i,k = ci,j = β. Two adjacent variables may be eliminated by the colour-

exchange operation. For example, if the two coloured edges are variables ej,i = (α, β) andei,k = (α, γ ), then the colour exchange ej,i ⊗ ei,k = (α, β) ⊗ (α, γ ) ⇒ (α, α) ◦ (β, γ ) can elimi-nate one of these variables. On the other hand, colour exchanges between two adjacent constantedges may introduce new variables. A colour-exchange operation is effective if it does not increasethe number of variables nc.

The Kempe walk of a (α, β) variable on a (α, β) path is a sequence of effective colour-exchangeoperations performed on its interior chain. Examples of variable eliminations by Kempe walksare provided in Figure 2. Consider the (r, b) path (b, r) ◦ (b, b) ◦ (r, b) shown in Figure 2(a); thevariable e1 = (b, r) can walk to another variable e2 = (r, b) by the following sequence of colour

(a)

(b)

(c)

Figure 2. Variable eliminations by Kempe walks (colour online only).

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Table 2. One-step move of (α, β) variable on (α, β) path.

Case Next step Operation Result

KW1 (α, β) ◦ (α, α) (α, β) ⊗ (α, α) ⇒ (α, α) ◦ (β, α) Step forwardKW2 (α, β) ◦ (α, β) (α, β) ⊗ (α, β) ⇒ (α, α) ◦ (β, β) Eliminate two variablesKW3 (α, β) ◦ (α, γ ) (α, β) ⊗ (α, γ ) ⇒ (α, α) ◦ (β, γ ) Eliminate one variableKW4 (α, β) ◦ (∅, ∅) (α, β) ⊗ (∅, ∅) ⇒ (α, α) Eliminate one variable

exchanges performed on its interior chain:

(b, r) ⊗ (b, b) ◦ (r, b) ⇒ (b, b) ◦ (r, b) ⊗ (r, b) ⇒ (b, b) ◦ (r, r) ◦ (b, b),

in which two variables are eliminated by colour exchanges.In a regular graph G, walks on an open (α, β) path always terminate on fictitious vertices at

both ends. If G is an irregular graph, then the degrees of some vertices are less than �, and a walkon an open (α, β) path may terminate on a vertex vi ∈ V(G) with a missing α or β edge. Sincewe have � colours available, a vertex with m < � has the freedom to choose any m out of �

colours. Thus, any missing colour at a vertex can be regarded as the ‘colour’ of some do not careedge, denoted as (∅, ∅). At those degenerate vertices, the colour-exchange operation involving ado not care edge is symbolically expressed as (α, β) ⊗ (∅, ∅) ⇒ (α, α). The examples depictedin Figures 2(b) and (c) show that a (b, r) variable on an open (b, r) path can be eliminated bywalking to either end of the path, regardless if it is a vertex or a fictitious vertex.

Table 2 summarizes all possible one-step Kempe walks of a (α, β) variable. Note that casesKW3 and KW4 only occur at the end of an open (α, β) path, and the number of variables nc

monotonically decreases as long as the (α, β) variable is walking on a (α, β) path.The negation of a coloured variable (α, β) is defined by −(α, β) = (β, α), referred to as the

colour vector in the opposite direction of (α, β). The following colour-inversion operation canchange the direction of a (α, β) variable contained in a maximum (α, β) path.

Definition 2.5 The colour inversion performed on a maximum (α, β) path H exchanges colourα and β on all links of H.

The colour inversion requires a sequence of colour exchanges involving all vertices of theinterior chain of the (α, β) path. It can only apply to a maximum (α, β) path, either a (α, β) cycleor an open (α, β) path, to avoid increasing the number of variables nc. For consistency, we alsouse the notation −(α, α) = (α, α) to denote the negation of a constant edge. It is interesting tonote that the colour-exchange operation of complex colours follows the same multiplication rulesas quaternion. This analogy is discussed in Appendix 1.

2.2 Canonical configurations

Kempe walks provide the most efficient and natural way to eliminate variables. Almost all vari-ables in the initial colour configuration can be eliminated by implementing Kempe walks. Aconfiguration is called a canonical configuration if no variables can be further reduced by a singlevariable Kempe walk. In a canonical configuration, it is easy to show from Lemma 2.3 that allremaining variables are contained in odd cycles, and every odd cycle contains exactly one singlevariable. Algorithm 1 exhaustively eliminates variables by Kempe walks.

Two canonical configurations of the Tutte graph resulted from the WKP algorithm are depictedin Figure 3, where green-coloured edges and fictitious vertices are faded to highlight (b, r) cycles.The canonical configuration shown in Figure 3(a) has two remaining (b, r) variables contained in

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Algorithm 1 Walk-on-Kempe-Path (WKP) algorithm

1 VariableList ← find all variables2 for each variable e = (α, β) in VariableList do3 Staring from variable e = (α, β), search for (α, β) path until

Case (1): find another variable or don’t care edge and form a (α, β) pathCase (2): return to the variable e = (α, β) and form a (α, β) cycle

4 For Case(1), move the variable e = (α, β) on the (α, β) path to eliminate variable(s) andupdate VariableList, then go to step 2

5 For Case(2), repeat step 3 with the next variable in the VariableList6 end for7 If the VariableList is empty then return a properly coloured graph8 If all variables in the VariableList end up with Case (2) then return a canonical configuration

(a) (b)

Figure 3. 3-edge-coloured Tutte graph (colour online only).

two disjoint odd (b, r) cycles. Figure 3(b) shows that a properly coloured Tutte graph has threeeven (b, r) cycles.

The performance of theWKP algorithm can be analysed with the assumption that the input graphis randomly generated. There are two conventional random graph models, namely the uniformrandom graph (also called Erdös–Rényi random graph) and the Bernoulli random graph. The setof uniform random graphs with n vertices and m edges is the equiprobable space of all n-vertex,m-edge-labelled simple graphs, while the set of n-vertex Bernoulli random graphs consists of alllabelled simple graphs in which each of the

(n2

)possible edges occurs with a probability p.

Let R�,n denote the set of �-regular uniform random graphs with n vertices; a subset of then-vertex uniform random graph space with a regular degree � and a fixed number of edgesm = �n/2. The performance of the WKP algorithm with an input graph selected from R�,n isgiven in the following theorem.

Theorem 2.6 For a �-regular graph G = (V , E) ∈ R�,n, the algorithm WKP either returns aproper �-edge colouring or a canonical configuration. The running time of the algorithm is onthe order of O(|V ||E|2).

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Proof According to Lemma 2.3, the variable ei,j = (α, β) must be the only variable contained inan odd (α, β) cycle if case (2) occurs in steps 3–5. The algorithm returns a properly coloured graphin step 7 if the VariableList is empty. Otherwise, every variable in the VariableList is containedin an odd cycle, and the algorithm returns a canonical configuration at step 8.

Next, the order of running time O(|V ||E|2) can be estimated by the running time of steps 3–5.For each variable selected from the VariableList, the running time of steps 3–5 is on the orderof O(|V |), because both the running time of path searching in step 3 and the number of colour-exchange operations performed in step 4 are bounded by the number of vertices |V |. Let (nc)

denote the number of times that steps 3–5 is executed, which is a function of the number of variablesnc in the initial configuration. Since at least one variable is eliminated in an updated VariableListwhen the loop (steps 3–5) repeats, (nc) ≤ ∑nc

i=0(nc − i) = O(nc2) = O(|E|2). Therefore, the

running time of the WKP algorithm is on the order of O(|V ||E|2). �

Let B�,n ⊂ R�,n denote the set of �-regular uniform random bipartite graphs with n vertices.If input to the WKP algorithm is a bipartite graph selected from B�,n, an immediate consequenceof the above theorem is the following corollary.

Corollary 2.7 For bipartite graph G = (V , E) ∈ B�,n, the WKP algorithm returns a proper�-edge colouring in O(|E| log |V |) time.

Proof Since there are no odd cycles in bipartite graphs, then step 3 always eliminates at leastone variable and loops at most nc times, where nc is the initial number of variables. Let Li be thelength of the search path of the ith input variable in step 3. Since the number of colour exchangesand the path search time are proportional to the total path length

∑nci=1 Li, the average path length

in a random graph is on the order of O(log |V |) [11]. Thus, the running time of the WKP algorithmfor a bipartite graph G = (V , E) can be estimated as follows:

nc∑i=1

Li = nc

∑nci=1 Li

nc= ncO(log |V |) = O(|E| log |V |). �

The simulation results presented in the next subsection show that the WKP algorithm actuallyperforms better than the above bound, which is a conservative estimate because we did not takethe trade-off between the number of variables nc and the path lengths in a bipartite graph intoconsideration.

2.3 Experiment results

The performance of the WKP algorithm is also verified by conducting experiments with randomlygenerated graphs. The random graph generator is based on an algorithm described in [19], whichuniformly produces �-regular random graphs with n vertices and m = �n/2 edges. Thus, theinput graph is randomly chosen from the set of R�,n with equal probability for a given number ofvertices n and degree �.

Experiments were conducted for two sets of randomly generated graphs: Set 1 contains the setof general �-regular random graphs R�,n and Set 2 only contains the set of �-regular bipartitegraphs B�,n. In each set, 100 graphs were randomly generated for every pair of (�, n), wheren = 100, 200, . . . , 10000 and � = 3, 6, 9.

The average number of steps required for running theWKP algorithm with Set 1 and Set 2 graphsare, respectively, plotted in Figures 4(a) and (b). According to Theorem 2.6 and Corollary 2.7,the running time of the WKP algorithm is bounded by the order of O(�2n3) with input graphsselected from R�,n, and O(�n log n) with input bipartite graphs selected from B�,n. However,

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D = 3D = 6D = 9

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

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2.5

3

3.5

Number of vertices

Ave

rage

num

ber

of s

teps

nee

ded

for

WK

P a

lgor

ithm

(b)D = 3D = 6D = 9

Figure 4. Experimental results of the WKP algorithm on (a) �-regular random graphs with n vertices R�,n and (b)�-regular bipartite graphs with n vertices B�,n.

our simulation results show that the performance of the WKP algorithm is better than thesetheoretical predictions. For each set of random graphs under consideration in our experimentstudy, the running time of the algorithm is almost linear in n for a fixed � as shown in Figure 4.

3. RWs on graphs

Despite the fact that an overwhelming number of variables in the initial colour configuration canbe eliminated by Kempe walks, the graph G is still not properly coloured if some variables remaintrapped in odd Kempe cycles. The Kempe walks are limited to colour exchanges performed withinalternating paths, which are fixed two-colour subgraphs H ⊆ G in any given colour configuration.In this section, we first introduce non-Kempe moves on path involving more than two coloursto break odd cycles and then describe an RW algorithm to systematically eliminate remainingvariables in canonical configurations.

3.1 Non-Kempe walks

A move of a tagged variable e1 = (α, β) along a predetermined path is effective if either the taggedvariable can step forward or some variables can be eliminated along the way. If e2 = (γ , δ), whereγ �= α and δ �= β, is the next edge adjacent to e1 on the selected path, then it is necessary to changethe colours of e2, either γ or δ, such that the operation e1 ⊗ e2 yields an effective move of e1. Thatis, effective walks on selected paths are actually walks on a dynamically changed Kempe path.From the operations listed in Table A1, it is obvious that an effective move of a tagged variablee1 = (α, β) requires one of the following types of colour inversion on the next edge e2:

• α-type: e2 = (γ , δ) → e2 = (α, ∗),• β-type: e2 = (γ , δ) → e2 = (∗, β).

A colour inversion applied to e2 may become invalid if the operation also involves e1. Theintend move is blocked if no valid colour inversions exist. We consider the following non-Kempemoves in a canonical configuration:

Case DW1: e1 ◦ e2 = (α, β) ◦ (γ , δ) , where γ �= α, δ �= β and δ ∈ {α, γ }.(1.1) δ = α, α-type: inverse the (γ , α) cycle H shown in Figure 5(a) that contains e2 = (γ , α);

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International Journal of Computer Mathematics 237

(a) (b) (c)

Figure 5. Illustration of non-Kempe walk Case DW1 (colour online only).

(a) (b)

Figure 6. Illustration of non-Kempe walk Case DW2 (colour online only).

(1.2) δ = γ , α-type: inverse the maximum (γ , α) path H shown in Figure 5(b) that contains theedge e2 = (γ , γ ).

The variable e1 = (α, β) is blocked if v3 = v0. Note that the variable e1 is not blocked ifv4 = v0, as illustrated in Figure 5(c), and the following sequence of operations can movee1 one step forward:• exchange colour on the interior chain from v2 to v0; hence, e1 = (γ , β) and e2 = (γ , α);• e1 ⊗ e2 = (γ , β) ⊗ (γ , α) ⇒ (γ , γ ) ◦ (β, α).

Case DW2: e1 ◦ e2 = (α, β) ◦ (γ , δ) , where γ �= α, δ �= β and δ /∈ {α, γ }.(2.1) α-type: inverse the maximum (γ , α) path H1 shown in Figure 6(a) that contains the vertex

v1 or(2.2) β-type: inverse the maximum (δ, β) path H2 shown in Figure 6(a) that contains the vertex

v2.

The variable e1 = (α, β) is blocked if v3 = v0 and v4 = v1, as illustrated in Figure 6(b).Table 3 lists all possible one-step non-Kempe moves of the tagged variable e1 = (α, β). It

should be noted that non-Kempe moves cannot avoid the possibility of blocking; otherwise,

Table 3. One-step non-Kempe moves of (α, β) variable on directional path.

Case Next step Colour inversion Operation Result

DW1.1 (α, β) ◦ (γ , α) (γ , α) → (α, γ ) (α, β) ⊗ (α, γ ) ⇒ (α, α) ◦ (β, γ ) Eliminate one variableDW1.2 (α, β) ◦ (γ , γ ) (γ , γ ) → (α, α) (α, β) ⊗ (α, α) ⇒ (α, α) ◦ (β, α) Step forwardBlocked DW1.2 If v3 = v0 BlockedDW2.1 (α, β) ◦ (γ , δ) (γ , δ) → (α, δ) (α, β) ⊗ (α, δ) ⇒ (α, α) ◦ (β, δ) Eliminate one variableDW2.2 (α, β) ◦ (γ , δ) (γ , δ) → (γ , β) (α, β) ⊗ (γ , β) ⇒ (α, γ ) ◦ (β, β) Eliminate one variableBlocked DW2 If v3 = v0 and v4 = v1 Blocked

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238 T.T. Lee et al.

variables can all be eliminated by walking them to a common vertex, resulting in a proper �-edge-coloured graph. The Petersen graph is a well-known counterexample to show that this isimpossible. A detailed discussion of blocked non-Kempe moves on the Petersen graph is providedin Appendix 1.

3.2 Colouring by RWs

An RW algorithm to systematically eliminate remaining variables is described in this subsec-tion. The algorithm starts with an arbitrary colour configuration of G with a set of � coloursC = {c1, c2, . . . , c�}. In a canonical configuration produced by the WKP algorithm described inSection 2, we first eliminate variables that contain colour c1, then eliminate the remaining variablesthat contain colour c2 and so on. The detail of the RW algorithm is listed in Algorithm 2.

Algorithm 2 The RW algorithm

1 Execute WKP algorithm with input graph G2 for each colour ci in C = {c1, c2, . . . , c�} do3 ci-VariableList ← find all variable edges containing colour ci

4 ni ← the initial number of variables in ci-VariableList5 for each variable e in ci-VariableList do6 k ← the number of remaining variables in ci-VariableList7 loop r(ni, k) times8 traverse a random adjacent edge e∗, repeat with another random adjacent edge e∗ in case of

blocking until the move e → e∗ succeeds9 search for a new Kempe path starting from e∗, check if any variable(s) can be eliminated by

Kempe walk of e∗ � Comment: the same Kempe path searching executed in the WKP algorithm10 if either step 8 or step 9 eliminates variable(s) containing ci then update ci-VariableList and

go to step 511 end loop12 end for13 if the random walk of every variable in ci-VariableList fails to eliminate any variable in r(ni, k)

steps then return χe(G) = � + 114 end for15 return proper �-edge-colouring of G

In the ith iteration started at step 2, the VariableList includes ni variables that contain colour ci,sometime called ci-variables. In steps 6–11, each selected ci-variable randomly moves at mostr(ni, k) − 1 steps, where the parameter r(ni, k) is a function to be determined later. Since a blockedvariable e = (α, β) can always find one of the two neighbouring α or β links, there is no riskof running into an infinite loop in step 8. Each successful move e → e∗ will change the graphcolouring to a new configuration, in which a Kempe path might have been created. Similar tosteps 3–5 of the WKP algorithm, a procedure of Kempe path searching is executed in step 9,and variables could be eliminated by the Kempe walk of e∗ on the newly created Kempe path.The algorithm repeats the loop (steps 6–11) with an updated ci-VariableList if any ci-variable iseliminated in step 9.

The problem is solved if all variables are eliminated and proper �-edge colouring is reached.On the other hand, if all ci-variables in the ith iteration fail to eliminate any variables withinr(ni, k) steps, for i ∈ {1, 2, . . . , �}, then the algorithm returns χe(G) = � + 1 in step 13.

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International Journal of Computer Mathematics 239

3.3 Analysis of randomized algorithm

Variables contained in odd cycles in a canonical configuration can only be eliminated by breakingodd cycles and hitting another variable. The probability of eliminating a variable is increasing withrespect to the allowed number of moves r(ni, k). Analysis of RWs on graphs usually considers awalk that starts at a vertex v ∈ V of a graph G = (V , E), and whenever it reaches any vertex u ∈ V ,choose an edge at random from those edges incident to u and traverse it. Our RW algorithm isimplemented on the line graph G = (V∗, E) induced from graph G = (V , E). The set of verticesV∗ is the set of fictitious vertices of the incidence graph G∗ = (V , E) and an ‘edge’ connectingtwo ‘vertices’ e∗

i , e∗j ∈ V∗ if and only if ei and ej are incident to the same vertex in G.

The performance of the RW algorithm is mainly determined by the hitting time of variables.Let Ak = ε1, ε2, . . . , εk be the set of ci-variables in the VariableList of ith iteration of the RWalgorithm. Suppose the variable εj takes T (k)

j steps to hit the odd cycle of another variable in theset Ak \ {εj}. Under the assumption that the input graph G = (V , E) is an Erdös–Rényi randomgraph, the hitting time, denoted as E[T (k)

j ] = h(n, k), is a function of the number of vertices n andthe number of remaining ci-variables k. If the input random graph G is �-edge colourable, thenthe following theorem shows that the RW algorithm with the parameter

r(ni, k) = ��nnih(h, k)� (1)

will return a proper �-edge colouring of G with a good probability.

Theorem 3.1 If input to the RW algorithm is a �-edge colourable Erdös–Rényi random graphG = (V , E), then the algorithm with parameter r(ni, k) will return a proper �-edge colouring ofG with a probability 1 − 1/n.

Proof We first prove that in the ith iteration of the algorithm

Pr{All ci-variables are eliminated} > 1 − 1

�n.

Let Ak = {ε1, ε2, . . . , εk} be the set of k remaining ci-variables in the VariableList. Suppose thatthe variable εj takes T (k)

j steps to hit another variable in the set Ak \ {εj}. Since the input to the

algorithm is an Erdös–Rényi random graph G = (V , E), it follows that T (k)

k , for j = 1, 2, . . . , k,are i.i.d. random variables. The probability that no vertices in the set Ak can be eliminated isgiven by

Pr{Ak fails to reduce to Ak−1} = Pr{T (k)1 ≥ r(ni, k), . . . , T (k)

k ≥ r(ni, k)}= (Pr{T (k)

j ≥ r(ni, k)})k . (2)

From Markov inequality and Equation (1), we have

Pr({T (k)j ≥ r(ni, k)})k ≤

(E[T (k)

j ]r(ni, k)

)k

=(

h(n, k)

r(ni, k)

)k

<

(1

�nni

)k

. (3)

The probability that all ci-variables can be eliminated in the ith iteration of the algorithm is givenby

Pr{all ci-variables are eliminated} >

ni∏k=1

(1 − Pr{Ak fails to reduce to Ak−1}). (4)

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Substituting Equations (2) and (3) into Equation (4), we have

Pr{all ci-variables are eliminated} >

ni∏k=1

(1 −

(1

�nni

)k)

>

ni∏k=1

(1 −

(1

�nni

))

=(

1 − 1

�nni

)ni

∼ 1 − 1

�n.

If χe(G) = �, then the probability that the algorithm returns a proper �-edge colouring of G isgiven by

Pr{output a proper �-edge colouring|χe(G) = �} =�∏

i=1

Pr{all ci-variables are eliminated}

>

(1 − 1

�n

)�

∼ 1 − 1

n. �

For �-regular random graph, an O(n3) upper bound on the hitting time was first obtained byAleliunas et al. [3]. Later, it was proved in [17] that the hitting time is at most 2n2 for a regulargraph. Let A ⊂ V be a proper subset of vertices and let v ∈ Ac, where Ac = V \ A; the hitting timeH[v, A] is the expected number of steps before any vertex u ∈ A is visited, starting from vertex v.For RW on a regular graph G = (V∗, E), it is proved in [2] that

H[v, A] < 4|Ac|2.

If input graph G = (V , E) of our RW algorithm is a �-regular Erdös–Rényi random graph, thehitting time E[T (k)

j ] = h(n, k) in the line graph G = (V∗, E) of G is the mean number of movesto hit the odd cycle of another variable, which should be bounded by

h(n, k) ≤ 4(|E| − k + 1)2 = 4

(�n

2− k + 1

)2

.

It follows that the total number of moves performed by the RW algorithm with parameterr(ni, k) is bounded by

≤�∑

i=1

ni∑k=1

kr(ni, k) ≤�∑

i=1

ni∑k=1

k�nnih(n, k) = 4�n�∑

i=1

ni

ni∑k=1

k(|E| − k + 1)2

≤ (�n)3�∑

i=1

ni

ni∑k=1

k ≤ (�n)3

2

�∑i=1

(n3i + n2

i ). (5)

Suppose the number of remaining variables in the canonical configuration is n0, then we have∑�i=1 ni ≤ n0 because the number of variables is monotonically decreasing throughout the entire

process. It follows that

≤ (�n)3

2

�∑i=1

(n3i + n2

i ) ≤ (�n)3

2(n3

0 + n20).

Hence, the total number of moves is bounded by = O(n30�

3n3). Furthermore, since the runningtime of step 8 in the loop from steps 6 to 11 is bounded by O(�|V |), we know from the proof

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International Journal of Computer Mathematics 241

of Theorem 2.6 that the running time of step 9 is on the order of O(|V |). Therefore, the timecomplexity of the RW algorithm with parameter r(ni, k) is on the order of O(n3

0�4n4). The

simulation results presented in the next subsection show that the RW algorithm with randominput graphs can perform much better than this bound.

3.4 Experiment results

Experiments were also conducted to verify the performance of the RW algorithm with randomlygenerated graphs. As we discussed in Section 2, the WKP algorithm with general input graphusually returns a canonical configuration with n0 remaining variables blocked in odd cycles. Thevalue for each pair (�, n) displayed in Figure 7 is the average number of remaining variables left inthe canonical configurations of 300 random graphs input to the WKP algorithm. As we expected,the number of remaining variables n0 is increasing with respect to the degree �; however, it isinsensitive to the number of vertices n of the input graph.

Almost all randomly generated graphs are �-colourable. The number of steps required by theRW algorithm to completely eliminate the remaining variables is plotted in Figure 8, in which thevalue corresponding to each pair of (�, n) is the average of 300 samples. Again, the simulationresults show that the running time of the RW algorithm is almost linear in n for fixed �. Thispoint is further confirmed by the percentage breakdown of the number of steps required by the300 random regular graphs with � = 9, for n = 100, 200, . . . , 10, 000, as shown in Figure 9.

The experimental results indicate that our RW algorithm performs well on average if the inputis a uniform random graph; however, the performance of the RW algorithm can be very poorif input to the algorithm is not a random graph. Holyer proved that 3SAT can be reduced toedge colouring of cubic graphs in polynomial time [15]. But the cubic graph G transformed from

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Number of vertices

Ave

rage

num

ber

of v

aria

bles

afte

r in

itial

run

of W

KP

alg

orith

m

D = 3D = 6D = 9

Figure 7. Average number of remaining variables in canonical configurations produced by WKP algorithm.

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1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

Number of vertices

Ave

rage

num

ber

of s

teps

nee

ded

for

RW

alg

orith

m

×104

D = 3D = 6D = 9

D = 3D = 6D = 9

Figure 8. Experiment results of RW algorithm.

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7

Number of vertices

Num

ber

of s

teps

nee

d fo

r W

KP

and

RW

alg

orith

m

95%75%25%Average

×105

Figure 9. Percentage breakdown of the running time for random regular graph with � = 9.

a 3SAT expression is far from random. Actually, the graph G is systematically constructed byinterconnecting two types of gadgets, each of them possesses a particular topological structure.Our experimental results show that the performances of the RW algorithm for those cubic graphsare substantially deviated from theoretical expectations.

An efficient variable elimination procedure for graphs with particular topological structuresmay require walks on specific directional paths other than randomly selected paths. For example,a spanning tree constructed from a specific input graph G may provide effective directional pathsthat guide remaining variables in canonical configurations to walk to the root of the tree. Intuitively,if variables can freely walk on those directional paths, then they will either intercept each otheron the way or eventually meet at the root. A blocked variable can be randomly deflected toanother nearby directional path and resume the walking towards the root in its remaining journey.However, the effectiveness of these variable elimination algorithms other than RWs depend onthe construction of directional paths embedded in the input graph G, which could be a challengingresearch topic of its own right in the future.

4. Conclusions

In this paper, edge colouring of simple graphs is solved by a variable elimination process similarto the solving of linear equations. The connections between graphs and linear equations providecornerstones in many areas such as electric circuit theory and Markov chains. In edge colouring of

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simple graphs, variables are eliminated by colour-exchange operations implemented on graphs.The problem is solved by a sequence of configuration transformations in the same manner assolving the puzzle of Rubik’s cube, which has a final configuration that can always be reached fromany initial configuration. In the case of edge colouring of graphs, however, only �-edge-colourablegraphs have final configurations.

Another related problem that could be solved by colour exchanges is finding the Hamiltoniancycles. A simple graph G may have more than one proper colour configurations. Consider the setof all proper colour configurations of a graph G as the state space of a Markov chain, in whicha state is Hamiltonian if it contains a two-coloured Hamiltonian cycle, then a solution could bereached by RWs on the Markov chain starting from an arbitrary canonical configuration. Thus,the application of the algebraic method proposed in this paper to graph factors and Hamiltoniancycles could be challenging research topics in the future.

Acknowledgements

This work was supported in part by Hong Kong RGC funded project 414012, in part by the National Science Foundationof China (61001074, 61172065, and 60825103), in part by the Qualcomm Corporation Foundation, and in part by 973program (2010CB328205, 2010CB328204).

References

[1] G. Aggarwal, R. Motwani, D. Shah, and A. Zhu, Switch scheduling via randomized edge coloring, Proceedings ofthe 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, p. 502.

[2] D. Aldous and J. Fill, Reversible Markov Chains and Random Walks on Graphs (monograph in preparation), 2002.Available at http://stat-www.berkeley.edu/users/aldous/RWG/book.html.

[3] R. Aleliunas, R. Karp, R. Lipton, L. Lovász, and C. Rackoff, Random walks, universal travelling, sequences, and thecomplexity of maze problems, Proceeding of 20th Annual Symposium on Foundations of Computer Science, 1979,pp. 218–223.

[4] K. Appel and W. Haken, Every Planar Map is Four Colorable, American Mathematical Society, Providence, RI,1989.

[5] R. Beigel and D. Eppstein, 3-coloring in time O(1.3446n): A no-MIS algorithm, 36th IEEE Symposium onFoundations of Computer Science, 1995, p. 444.

[6] R. Cole, K. Ost, and S. Schirra, Edge-coloring bipartite multigraphs in O (E log D) time, Combinatorica 21 (2001),pp. 5–12.

[7] J. Conway and D. Smith, On Quaternions and Octonions, AK Peters, Natick, MA, 2003.[8] D. Dubhashi, D. Grable, and A. Panconesi, Near-optimal, distributed edge colouring via the nibble method, Theoret.

Comput. Sci. 203 (1998), pp. 225–251.[9] J. Edmonds, Paths, trees, and flowers, in Classic Papers in Combinatorics, I. Gessel and G.-C. Rota, eds., Birkhäuser,

Boston, 1987, pp. 361–379.[10] D. Eppstein, Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction, Proceedings of the

Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, 2001, pp. 329–337.[11] A. Fronczak, P. Fronczak, and J. Hołyst, Average path length in random networks, Phys. Rev. E 70(5) (2004), Article

ID 056110. Available at http://pre.aps.org/abstract/PRE/v70/i5/e056110.[12] H. Gabow, T. Nishizeki, O. Kariv, D. Leven, and O. Terada, Algorithms for edge-coloring graphs, Tech.

Rep., 1985.[13] D. Grable and A. Panconesi, Nearly optimal distributed edge colouring in O (log log n) rounds, Proceedings of the

Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 1997, pp. 278–285.[14] M. Hilgemeier, N. Drechsler, and R. Drechsler, Fast heuristics for the edge coloring of large graphs, Proceedings

of the Euromicro Symposium on Digital System Design, 2003, pp. 230–237.[15] I. Holyer, The NP-completeness of edge-colouring, SIAM J. Comput. 10 (1981), pp. 718–720.[16] A. Kempe, On the geographical problem of the four colours, Am. J. Math. 2 (1879), pp. 193–200.[17] L. Lovász, Random walks on graphs: A survey, in Combinatorics, Paul Erdos is Eighty, Vol. 2, D. Miklós, V. T. Sós,

and T. Szonyi, eds., János Bolyai Mathematical Society, Budapest, Hungary, 1993, pp. 1–46.[18] N. Robertson, D. Sanders, P. Seymour, and R. Thomas, The four-colour theorem, J. Combin. Theory Ser. B 70

(1997), pp. 2–44.[19] A. Steger and N. Wormald, Generating random regular graphs quickly, Combin. Probab. Comput. 8 (1999),

pp. 377–396.[20] P. Tait, Note on a theorem in geometry of position, Trans. R. Soc. Edinburgh 29 (1880), pp. 657–660.[21] D. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 2001.

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Table A1. Correspondence between quaternion multiplication and colour-exchange operations.

Correspondence

A = (α, α) = −(α, α), B = (β, β) = −(β, β), C = (γ , γ ) = −(γ , γ ) → 1(α, β) → i, (β, α) → −i, (β, γ ) → j, (γ , β) → −j, (γ , α) → k, (α, γ ) → −k

Quaternion multiplication Colour-exchange operation

× −i −j −k ⊗ (β, α) (γ , β) (α, γ )

i 1 −k j (α, β) −AB (α, γ )B A(β, γ )j k 1 −i (β, γ ) B(γ , α) −BC (β, α)Ck −j i 1 (γ , α) (γ , β)A C(α, β) −CA

Appendix 1. Colour quaternion

The concept of quaternion was introduced by Hamilton in 1843 as an extension of complex numbers [7]. The basiselements of quaternions are customarily denoted as 1, i, j, and k. The correspondences between quaternion multiplicationand colour-exchange operations are summarized in Table A1. Note that the minus sign of diagonal entries in the colour-exchange table indicates the necessary colour inversion of the right operand due to the consistency requirement at eachvertex of the colour configuration. The exchange operation listed in Table A1 implies that if two adjacent variables haveone colour in common, then at least one of them can be eliminated. This is an important property in the construction ofedge-colouring algorithms for eliminating variables involving more than three colours.

Appendix 2. Snarks

The main difference between solving linear equations and edge colouring is the recognition of the final state. The inconsis-tency of a system of linear equations can be easily identified by variable eliminations in polynomial time. But eliminatingvariables in a class 2 graph may run into an infinite loop. The smallest class 2 cubic graph, called snark, is Petersengraph. The three-colour canonical configuration of the Petersen graph shown in Figure A1(a) contains two variables intwo disjoint odd cycles. The two odd cycles behave the same as two parallel lines in a Euclidean space; they can nevercross each other, which is the geometric interpretation of inconsistent linear equations.

In general, all three-colour canonical configurations of Petersen graph are isomorphic. That is, corresponding to anymaximum (α, β) path H1 in a canonical configuration φ1, there is a maximum (α, β) path H2 in another canonicalconfiguration φ2, such that the two subgraphs H1 and H2 are graph isomorphic [21]. This is the reason that any variableelimination procedures can never halt when all canonical configurations of graph G are isomorphic.

The configuration shown in Figure A1(b) is the same as that in Figure A1(a), but the two disjoint (b, g) cycles areseparated in the plane and connected by constant (r, r) edges. In Figure A1(b), we consider the walk on the shortestdirectional path e6,7 ◦ e3,6 ◦ e2,3 = (b, g) ◦ (r, r) ◦ (b, g) between the two variables e6,7 and e2,3. Both moves, e6,7 → e3,6and e2,3 → e3,6, are blocked according to the blocked move of DW1.2 described in Table 3. In fact, walks on any shortestdirectional path will be blocked if the two variables are directly connected by a constant (r, r) edge.

Figure A1(c) shows that the variable e6,7 successfully walks to edge e5,7. However, the resulting configuration is thesame as the previous one up to some permutation of vertices. The correspondence between the vertices in Figure A1(b)

(a) (b) (c) (d)

Figure A1. Isomorphic configurations of Petersen graph (red colour is faded). (a) Canonical configuration; (b) twin-cycleconfiguration, e6,7 → e3,6 and e2,3 → e3,6 are blocked; (c) e6,7 → e5,7 succeed; and (d) twin-cycle configuration after themove e6,7 → e5,7 (colour online only).

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International Journal of Computer Mathematics 245

and (c) is shown in Figure A1(d). Therefore, whenever a blocked variable walks out of the odd cycle, the new configurationis isomorphic to the previous one.

The edge colouring of general graphs also faces the halting problem. In fact, any �-regular graph G with an odd numberof vertices is a class 2 graph. A simple example is K5, where n = 5, � = 4, and χe(K5) = 5. In a four-colour canonicalconfiguration of K5, two distinct variables are mutually blocked and can never be eliminated, which corresponds to theblocked move of DW2.2 given in Table 3.

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