On the Maximum Degree of 3t-Critical GraphsBarioli, van der Merwe Max Degree Introduction Background...

IntroductionBackground

General ResultsSharp Upper and Lower Bounds

Summary

On the Maximum Degree of 3t-Critical Graphs

Francesco Barioli Lucas van der Merwe1

Department of MathematicsUniversity of Tennessee at Chattanooga

1Research supported by Math. Depart. UTCBarioli, van der Merwe Max Degree



Summary

Outline

1 Introduction

2 Background

3 General ResultsGeneral Bounds

4 Sharp Upper and Lower Boundsα(G) = 2Crown GraphsClaw-Free Graphs

5 Summary

Barioli, van der Merwe Max Degree



Summary

N(u) is the open neighborhood of a vertex u in a graph,whereas N[u] = N(u) ∪ {u} is the closed neighborhood ofu.The degree of a vertex u is |N(u)| denoted deg(u).The minimum degree and maximum degree of G aredefined by

δ(G) = minv∈V (G)

deg(v), ∆(G) = maxv∈V (G)

deg(v),

respectively.




Summary

S is a total dominating set of G if every vertex of V (G) isadjacent to some vertex of S. We write S �t G. (Firstintroduced by Cockayne, Dawes and Hedetniemi, (1980)).

The minimum cardinality of S is the total domination numberof G denoted by γt (G).

S is a dominating set of V (G) if every vertex of V (G)− S isadjacent to some vertex of S.

The minimum cardinality of S is the domination number of Gdenoted by γ(G).




Summary

The total domination number deceases by at most twowhen adding an edge to a graph G. That is,

γt (G)− 2 6 γt (G + e) 6 γt (G)

Graphs for which γt (G + e) = γt (G)− 2 for each e /∈ E(G)are called supercritical.




Summary

If for three given vertices u, v , w we have {u, v} �t V (G)− w ,we write uv 7→ w .

Theorem 1Let γt (G) = 3. Then is G is 3t -critical if and only if for any pairof non adjacent vertices u and v, either(1) {u, v} � V (G), or(2) uw 7→ v for some w ∈ N(u), or(3) vw 7→ u for some w ∈ N(v).




Summary

1

2

3

4

5

6

7

{1,4,5} is a γt -set among others. Now add an edge e ∈ G.




Summary

1

2

3

4

5

6

7




Summary

Background

A graph G is diameter 2-critical if diam(G) = 2 anddiam(G − e) > 2 for any e ∈ E(G).

Hanson and Wang showed the following important relationship.

Theorem 2A graph is diameter 2-critical if and only if its complement iseither 3t -critical or 4t -supercritical.




Summary

Murty and Simon posed a conjecture on diameter 2-criticalgraphs, namely:

Conjecture 3If G is a diameter 2-critical graph with order n and size m, thenm ≤ dn2/4e, with equality if and only if G is the completebipartite graph Kd n

2 e,bn2 c

.

Theorem 4If G be a graph with order n and size m, then(1) if G is 3t -critical, then m > dn(n − 2)/4e;(2) if G is 4t -supercritical, then m ≥ dn(n − 2)/4e.

Condition (2) has been settled by Hanson and Wang (2003).




Summary

Recently, Condition (1) was settled for the two following familiesof 3t -critical graphs. (Henning, Haynes, Yeo, vdM)

Theorem 5If G is a 3t -critical graph of diameter 3, order n, and size m,then m > dn(n − 2)/4e.

Theorem 6If G is a 3t -critical claw-free graph of order n and size m, thenm > dn(n − 2)/4e.




Summary

General Bounds

Theorem 7Let G be a 3t -critical graph. Then |G| > 5. Furthermore, if|G| = 5 then G = C5, the cycle on five vertices.

Theorem 8(Cockayne et al .) If a graph G is connected and∆(G) < |G| − 1, then γt (G) 6 |G| −∆(G).

Observation 9

Any graph G with γt (G) = 3 has ∆(G) 6 |G| − 3. Moregenerally, ∆(G) ≤ |G| − γt (G).




Summary

General Bounds

Theorem 10The only 3t -critical |G| − 3-regular graph G is C5.

Theorem 11

Let G be a 3t -critical graph with |G| > 5. Then ∆(G) ≥ d12 |G|e.

Combining Observation 9 and Theorem 11, gives:

Theorem 12

If G is a 3t -critical graph, then d12 |G|e ≤ ∆(G) ≤ |G| − 3.




Summary

α(G) = 2Crown GraphsClaw-Free Graphs

The independence number α(G) of a graph G is defined as themaximum number of pairwise disjoint vertices of G.

Lemma 13If γt (G) = 3 and α(G) = 2, then G is 3t -critical.

Another straightforward yet important fact:

Lemma 14

Let G be a 3t -critical graph with α(G) = 2. Then two verticesare adjacent if and only if they share a common non-neighbor.

In particular, if u and v are non-adjacent vertices, we have

|G| = deg(u) + deg(v)− |N(u) ∩ N(v)|+ 2. (1)




Summary


Theorem 15

Let G be a 3t -critical graph with α(G) = 2. Then∆(G) > d3

5 |G| − 1e.

Outline of proof.Let n = |G|, k = ∆(G).Among all pairs of nonadjacent vertices in G, let (u, v) be apair maximizing deg(u) + deg(v).deg(u) + deg(v) is maximum when |N(u) ∩ N(v)| ismaximum.Partition V (G) \ {u, v} into the three subsets

A = N(u)\N(v), B = N(v)\N(u), C = N(u)∩N(v).




Summary


u v

A B

C

|A|+ |C| = deg(u) 6 k ; |B|+ |C| = deg(v) 6 k , (2)

By Lemma 14, both A and B are complete.




Summary


ab

c

u v

C is necessarily non-empty. {c, v} � V (G) \ A, while{c, v} 6� V (G). There exists a ∈ A that is not adjacent to c.Similarly, b ∈ B nonadjacent to c.




Summary


N (a)

av

∣N B(a )∣≤∣C∣

B

{a, c} � G and {b, c} � G. By the maximality ofdeg(u) + deg(v), deg(a) 6 deg(u). Since a � A \ {a1}, wehave

|NB(a)| = deg(a)− (|A| − 1)− 1− |NC(a)|

6 deg(a)− |A| 6 deg(u)− |A| = |C|.




Summary


N B(c)

v

∣N B(c )∣<∣C∣

c B

b

Since v ∈ N(b) ∩ N(c), we have NB(c) ( N(b) ∩ N(c), so that

|NB(c)| < |N(b) ∩ N(c)| 6 |N(u) ∩ N(v)| = |C|.




Summary


Finally, since {a, c} � B, we have

|B| 6 |NB(a)|+ |NB(c)| 6 2|C| − 1. (3)

From (2) and (3) we then obtain

n = |A|+ |B|+ |C|+ 2

=3 (|A|+ |C|) + 2 (|B|+ |C|) + |B| − 2|C|+ 6

3

65k + 5

3=

53

(k + 1),

which completes the proof.




Summary


Vertices u and v of G are twin vertices (or duplicatevertices) if N[u] = N[v ]. A graph G′ is said to be obtainablefrom G by vertex duplication if G′ has twin vertices u and vsuch that G = G′ − {v}.

Lemma 16

Let G be a graph with α(G) = 2, and let G′ be obtainable fromG by vertex duplication. Then

i. α(G′) = 2;ii. G′ is 3t -critical if and only if G is 3t -critical.

It is important to observe that a similar result does not holdwhen α(G) > 2.




Summary


A graph G is said to be a 5-cycle-type graph if G can beobtained from the 5-cycle C5 through a sequence of vertexduplications. In particular, if the sizes of these cliques arec1, . . . , c5, we will write G = C(i1, i2, i3, i4, i5).

ss ssss

ss sJJ

Figure: The graph C(3,1,2,2,1).




Summary


Proposition 17

Let G = C(i1, i2, i3, i4, i5) be a 5-cycle-type graph. Then

i. |G| =5∑

j=1

ij ,

ii. ∆(G) = maxj

(ij−1 + ij + ij+1)− 1,

iii. α(G) = 2,iv. G is 3t -critical,

where indices have to be considered modulo 5.




Summary


Theorem 18

Let n, k ∈ N such that n > 5 and d3n5 − 1e 6 k 6 n − 3. Then

there exists a 3t -critical graph G with α(G) = 2, |G| = n, and∆(G) = k.

Proof. Using Proposition 17, it is easy to show thatG = C (n − 4p, bpc, dpe, dpe, bpc) where p = n−k−1

2 satisfies allof the requirements.




Summary


Another interesting property of 5-cycle-type graphs is that theyexhaust the class of 3t -critical graphs with α(G) = 2 and∆(G) = |G| − 3, as stated in the following theorem.

Theorem 19

Let G be a 3t -critical graph with α(G) = 2 and ∆(G) = |G| − 3.Then G is a 5-cycle-type graph.




Summary


Another interesting class of 3t -critical graphs withα(G) = 2, is the subclass of circulant graphsC(n) = Cin

(1,2, . . . , n−2

3

), for any n ≡ 2( mod 3).

We just observe that, since C(5) = C5, we can extend thisclass of circulant graphs by vertex duplication to containthe class of 5-cycle-type graphs. However, there are stillexamples of 3t -critical graphs G with α(G) = 2 that cannotbe obtained from any circulant graph of the form C(n)through vertex duplication, as, for instance, thecomplement of the Petersen graph.




Summary


s ss

ssss

s s sss

sssss

s

Figure: The graphs C(8) and the complement of Petersen graph




Summary


A graph G is said to be a Crown graph if G is 3t -critical,and, for any pair of nonadjacent vertices u and v in G,there exist vertices x and y in G such that ux 7→ v andvy 7→ u.

a2

a3

b2

b3

G+a2a3 : a2b3→a3 a3b2→a2

G :




Summary


We note that if G is a 3t -critical graph with nonadjacent verticesu and w in G, and uv 7→ w , for some vertex v , then we have

|G| = deg(u) + deg(v)− |N(u) ∩ N(v)|+ 1. (4)

N (u)∩N (v )

u v

wG : uv→w




Summary


Lemma 20

Let G be a Crown graph. Then |G| 6 ∆(G) + 12δ(G) + 2.

Proof.Let k = ∆(G), δ = δ(G), and let u be a vertex of degree δ.Let A = N(u), and let B = V (G)− (A ∪ {u}).Then |B| = n − δ − 1 with B = {b1, . . . ,bn−δ−1}. For eachbi ∈ B there is at least one distinct vertex ai ∈ A such thatuai 7→ bi .Let br and bs be vertices in B such that br bs 7→ u. Notethat uai 7→ bi implies that ai , i 6= r , s, is adjacent to both brand bs.




Summary


Thus, br and bs have at least n − δ − 3 common neighborsin A, that is |N(br ) ∩ N(bs)| > n − δ − 3.By (4) we have

n 6 deg(br ) + deg(bs)− (n − δ − 3) + 1 6 2k − n + δ + 4,

which yields n 6 k + 12δ + 2.




Summary


Since δ(G) 6 ∆(G), Lemma 20 can be adapted to provide abetter lower bound for ∆(G) in terms of |G| in the case ofCrown graphs.

Corollary 21

Let G be a Crown graph. Then ∆(G) >⌈2

3(|G| − 2)⌉.




Summary


When equality holds in Corollary 21, then Lemma 20 yieldsδ(G) = ∆(G) so that we have the following interesting result.

Corollary 22

Let G be a Crown graph with ∆(G) =⌈2

3(|G| − 2)⌉. Then G is

regular.

We also establish a lower bound for δ(G) in terms of |G|,namely:

Lemma 23

Let G be a Crown graph. Then δ(G) >⌈1

2(|G| − 1)⌉.




Summary


When, in Lemma 23, equality is attained, the result of Corollary21 can be further improved as follows.

Corollary 24

Let G be a Crown graph with δ(G) = 12(|G| − 1). Then

∆(G) >⌈

3|G|−74

⌉.

Proof. From Lemma 20 we have

|G| 6 ∆(G) +14

(|G| − 1) + 2

which is equivalent to ∆(G) > 3|G|−74 .




Summary


The upper bound provided by Proposition 9, |G| − 3, issharp within the class of Crown graphs, and for n arbitrarilylarge.

The lower bound provided in Corollary 21 is also sharp forn arbitrarily large. Indeed, it is sufficient to observe that thecirculant graphs C(n) defined previously are Crown graphswith |G| = n, ∆(G) = 2(n−2)

3 .




Summary


A graph G is said to be claw-free if it does not contain K1,3 asan induced subgraph. As 5-cycle-type graphs are claw-free, inview of Theorem 18 we do not expect claw-freeness to lead toany significant improvement in lower and upper bounds for∆(G), in general. However, as long as α(G) > 2, the lowerbound may be improved as shown in the following result.

Theorem 25

Let G be a 3t -critical claw-free graph, with α(G) > 2. Then∆(G) >

⌈23(|G| − 2)

⌉.




Summary


For claw-free graphs we have the following theorem, similar toTheorem 18 obtained in the case α(G) = 2.

Theorem 26

Let n, k ∈ N such that n > 8 and⌈2

3(n − 2)⌉6 k 6 n − 3. Then

there exists a 3t -critical claw-free graph G with α(G) > 2,|G| = n, and ∆(G) = k.




Summary

Summary

While for the case α(G) = 2, as well for Crown graphs and forclaw-free graphs, we could establish sharp lower bounds on∆(G), we still lack a sharp lower bound for the general case, aswe feel that the bound

⌈12 |G|

⌉provided in Theorem 11 is not

sharp, starting, probably, from n = 12.It is also somehow bizarre that in the general case, theadditional condition α(G) = 2 yields a tighter lower bound(Theorem 15), while, within the class of claw-free graphs, thecondition α(G) > 2 is the one that provides the tighter lowerbound (Theorem 25). This fact may be due, again, to the lack ofa sharp lower bound for the general case, which, at this pointwe really feel should be studied, and, possibly, determined.


On the Maximum Degree of 3t-Critical GraphsBarioli, van der Merwe Max Degree Introduction Background...

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