Graph Coloring prepared and Instructed by Shmuel Wimer
Graph Coloring prepared and Instructed by Shmuel Wimer Eng. Faculty, Bar-Ilan University March 2014 Graph Coloring 1
Vertex Coloring March 2014 Graph Coloring 2
Examples. Every bipartite graph is 2 -colorable. Every even cycle graph is 2 -colorable (it is bipartite). Every odd cycle graph is 3 -colorable and 3 -critical. 2 -colorability can be tested with BFS. (how? ) March 2014 Graph Coloring 3
Yes! March 2014 Graph Coloring 4
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Upper Bounds of Chromatic Number March 2014 Graph Coloring 6
Different orderings may yield smaller upper bounds. Finding the best ordering is hard. Example. Register allocation and interval graphs. Consider the logical registers used by a compiler, each has start and end time. What is the smallest number of physical registers that can be used? March 2014 Graph Coloring 7
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Coloring of Directed Graphs March 2014 Graph Coloring 12
2 4 3 1 6 1 March 2014 5 Graph Coloring 13
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Brooks’ Theorem March 2014 Graph Coloring 16
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2 nd case: G is not 2 -connected. March 2014 Graph Coloring 19
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Chromatic Polynomials We shall associate with any graph a function telling whether or not it is 4 -colorable. This study was motivated by the hope to prove the Four. Color Theorem, which by that time was a conjecture. March 2014 Graph Coloring 25
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Corollary. The chromatic function is a polynomial. March 2014 Graph Coloring 28
The process is finite. It ends with producing complete graphs, whose chromatic functions are polynomial. The chromatic function is therefore a finite sum of polynomials, which must be polynomial too. ■ Example. Scheduling feasibility. Lectures scheduling is in order, for which some time slots are given (e. g. campus is open). There is no limit on available rooms. March 2014 Graph Coloring 29
It is known that some lectures cannot take place in parallel (e. g. some students are registered to both). Is scheduling feasible? How many schedules there are? March 2014 Graph Coloring 30
Example. = = March 2014 + + + Graph Coloring + 31
= March 2014 + +2 Graph Coloring + 32
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Edge Coloring An edge coloring is proper if adjacent edges have different colors. All coloring henceforth are assumed proper. March 2014 Graph Coloring 34
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Edge Coloring of Bipartite Graphs March 2014 Graph Coloring 38
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Line Graphs Many questions about vertices have natural analogues involving edges. Independent sets have no pairs of adjacent vertices; matchings have no adjacent edges. Vertex coloring partitions the vertices into independent sets; edges can be partitioned into matching. March 2014 Graph Coloring 51
Line Graphs Characterization Since an edge connects two vertices, those vertices imply two cliques at most. March 2014 Graph Coloring 52
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impossible clique => clique Krausz’s theorem does not directly yield an efficient test for line graph, which the following does. March 2014 Graph Coloring 54
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- Slides: 55
