David Laughon CS594 Graph Theory Graph Coloring. Coloring – Assignment of labels to vertices...

Upload
whitneynewton 
Category
Documents

view
252 
download
12
Transcript of David Laughon CS594 Graph Theory Graph Coloring. Coloring – Assignment of labels to vertices...
David LaughonCS594 Graph Theory
Graph Coloring
• Coloring – Assignment of labels to vertices
• kcoloring – a coloring where
• Proper kcoloring – kcoloring where vertices have different labels if they are adjacent
• Chromatic number – least k for which G is kcolorable  χ(G)
Definitions
• A Graph is kchromatic if χ(G) = k
• Optimal coloring – proper kcoloring of a kchromatic graph– Vertexcoloring problems
• Is a graph kcolorable for given k?• What is χ(G) / what is the optimal coloring?
Definitions
• Fourcolor conjecture – Francis Guthrie, 1852 (F.G.)– Can any map be colored using at most 4 colors so that
adjacent regions are not the same color?• Many incomplete proofs (Kempe)
– “Counterexamples”• 5color theorem proved in 1890 (Heawood)• 4color theorem finally proved in 1977 (Appel, Haken)
– First major computerbased proof• Graph coloring applies to nonplanar graphs as well
History
• Martin Gardner, April 1975 edition of Scientific American
• As an April fool’s joke, claimed graph required 5 colors
4color “Counterexample”
Proper 6coloring Optimal 4coloring
Examples
• For complete graphs, χ(G) = n
• Each vertex has n1 edges that connect to every other vertex– Forces each vertex to
have a unique color
Examples – Kn
• A graph is 2colorable iff it is bipartite
Examples
• ω(G) – size of largest clique in G• χ(G) ≥ ω(G)
– Clique of size n requires n colors– Can be a tight bound, but not always
Examples
Examples
χ(G) = 7, ω(G) = 5
• Mycielski’s Construction– Can be used to make
graphs with arbitrarily large chromatic numbers, that do not contain K3 as a subgraph
Examples
• χ(G) ≤ Δ(G) + 1
• Greedy Algorithm:– Put the vertices of a graph in a sequence– For each vertex in the sequence, assign it the lowest
indexed color not already assigned to adjacent vertices• Not guaranteed to be optimal for every possible
sequence• Guaranteed optimal for at least one sequence
Greedy Coloring
Vertex 0 1 2 3 4 5
Color Yellow Yellow Yellow Green Green Green
Greedy Coloring Example
Vertex 0 3 1 4 2 5
Color Yellow Yellow Green Green Purple Purple
Greedy Coloring Example
• A path in a graph that alternates between 2 colors
• First used by Kempe in his incorrect proof of the 4color theorem
• Used in 5color theorem and 4color theorem proofs
Kempe Chains
• All planar graphs can be colored with at most 5 colors
• Basis step: True for n(G) ≤ 5• Induction step: n(g) > 5• There exists a vertex v in G of degree at most 5 (Theorem
6.1.23)• G – v must be 5colorable by induction hypothesis
5color theorem
• If G is 5colorable, done• If G is not 5 colorable, we
have:
• Is there a Kempe chain including v1 and v3?
5color Theorem
There is no Kempe chain There is a Kempe chain
5color Theorem
There cannot be a Kempe chain including v2 and v4
v4 cannot directly influence v2
5color Theorem
• Similar to vertex coloring, except edges are colored– Adjacent edges have different colors
Edge Coloring
• Every edgecoloring problem can be transformed into a vertexcoloring problem
• Coloring the edges of graph G is the same as coloring the vertices in L(G)
• Not every vertexcoloring problem can be transformed tin an edgecoloring problem– Every graph has a line graph, but not every graph is a
line graph of some other graph
Edge Coloring
K4 edgecoloring L(K4) vertexcoloring
Edge Coloring
• Each vertex in G has a positive integer label x(v): the number of colors that must be assigned to that vertex
• The color sets of adjacent vertices must be disjoint
Multicoloring
{Yellow, Green, Purple, Red}
{Blue}
{Yellow, Green}
{Blue}
χ(G) = 5
• Every multicoloring problem can be transformed to a vertexcoloring problem– For each vertex with x(v) = n, replace it with a clique of
size n.– Add an edge from each vertex in the new clique to
every vertex that the original vertex was adjacent to.– Single vertexcoloring now solves the problem
Multicoloring
Multicoloring
χ(G) = 5
• Scheduling• Register allocation• VLSI channel routing• Biological networks (Khor)• Testing printed circuit boards (Garey, Johnson, & Hing)• Sudoku
Applications
• Each cell is a vertex• Each integer label is a
“color”• A vertex is adjacent to
another vertex if one of the following hold:– Same row– Same column– Same 3x3 grid
• Vertexcoloring solves Sudoku
Applications: Sudoku
• Decide if a graph is kcolorable is NPcomplete• Determining χ(G) is NPhard• kcolorable – O(2.445^n) (Lawler)• 3colorable – O(1.3289^n) (Beigel, Eppstein)• 4colorable – O(1.7272^n) (Fomin, Gaspers, & Saurabh)• Alternative methods of solving graph coloring
– Swarm intelligence (Dorrigiv, Markib)
State of the Art
• Hadwiger Conjecture– Every kchromatic graph has a subgraph that becomes
Kk through edge contractions
– Open for k ≥ 7
Open Problems
• Erdős–Faber–Lovász conjecture– Consider k complete
graphs with exactly k vertices. If every pair of complete graphs shares at most one vertex, then the union of the graphs can be colored with k colors
Open Problems
• F. G. (June 10, 1854), "Tinting Maps", The Athenaeum: 726.• Heawood, P. J. (1890), "MapColour Theorems", Quarterly Journal of
Mathematics, Oxford 24: 332–338• Appel, Kenneth; Haken, Wolfgang (1977), "Every Planar Map is Four Colorable
Part I. Discharging", Illinois Journal of Mathematics 21: 429–490 • Appel, Kenneth; Haken, Wolfgang; Koch, John (1977), "Every Planar Map is
Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567
• Appel, Kenneth; Haken, Wolfgang (October 1977), "Solution of the Four Color Map Problem", Scientific American 237 (4): 108–121
• Khor, S., "Application of graph colouring to biological networks," Systems Biology, IET , vol.4, no.3, pp.185,192, May 2010
• Garey, M.R.; Johnson, D.; Hing So, "An application of graph coloring to printed circuit testing," Circuits and Systems, IEEE Transactions on , vol.23, no.10, pp.591,599, Oct 1976
References
• Lawler, E.L. (1976), "A note on the complexity of the chromatic number problem", Information Processing Letters 5 (3): 66–67
• Beigel, R.; Eppstein, D. (2005), "3coloring in time O(1.3289n)", Journal of Algorithms 54 (2)): 168–204
• Fomin, F.V.; Gaspers, S.; Saurabh, S. (2007), "Improved Exact Algorithms for Counting 3 and 4Colorings", Proc. 13th Annual International Conference, COCOON 2007, Lecture Notes in Computer Science 4598, Springer, pp. 65–74
• Dorrigiv, M.; Markib, H.Y., "Algorithms for the graph coloring problem based on swarm intelligence," Artificial Intelligence and Signal Processing (AISP), 2012 16th CSI International Symposium on , vol., no., pp.473,478, 23 May 2012
References
• Prove that every graph has a vertex ordering such that the greedy coloring algorithm produces an optimal coloring
• Given a kchromatic graph and an optimal coloring of it, prove that for each color i there is a vertex with color i that is adjacent to vertices of all the other k1 colors
Homework