Vertex Coloring Chromatic Number Conflict Resolution Conflict Scheduling Vertex Coloring Chromatic Number Conflict Resolution Mathematics in Management Science Spring 2015

Vertex Coloring Example

Vertex Coloring Color all vertices of graph so that any two vertices joined by an edge have different colors. The minimum number of colors needed is the chromatic number of the graph.

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Vertex Coloring Minimum number of colors needed (to have a vertex coloring) is the chromatic number of the graph. To see that a graph has chromatic number CN, must show: there is vtx coloring with NC colors, cannot color with less than NC colors.

Coloring Circuits L = # edges = # of vtxs . The length of a circuit is Every circuit can be colored using 2 or 3 colors. The chromatic number of a circuit is CN = 2 if even length, CN = 3 if odd length.

Coloring Complete Graphs A graph is complete if every pair of vtxs is joined by an edge. Any vertex coloring of a complete graph with N vertices must use N different colors. The chromatic number of KN is CN = N .

Useful Fact If graph has subgraph with chrom number N, then the bigger graph will have chrom number at least N. (Can’t use fewer colors!) This useful when a bigger graph has a smaller complete graph built into it.

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CN ≤ maximum vertex valence. Brooks’ Theorem G a graph which is not complete nor a circuit (of odd length) G’s chromatic number satisfies CN ≤ maximum vertex valence.

Chromatic Number Minimum number of colors need. The chromatic number of a cplt graph is CN = # of vtxs . The chromatic number of a circuit CN = 2 if even length CN = 3 if odd length All other graphs have CN at most the maximum vertex valence .

Chromatic Number Don’t have vtx coloring algorithms; just be smart. Important to“clean up” graph, so eaier to see what is happening. Start coloring most complicated parts; look for complete subgraphs. This tell us minimum size of our “palette” (lower estimate for CN).

Example What is the chromatic number of the following graph? A C B D E G F H (What is the table of conflicts?)

Observe that the graph contains a complete graph on 4 vertices: Example Observe that the graph contains a complete graph on 4 vertices: A C B D E G F H

Extend the 4 colors to the other complete graph: Example Extend the 4 colors to the other complete graph: A C B D E G F H

Pick colors for the remaining vertices from those already in use: Example Pick colors for the remaining vertices from those already in use: A C B D E G F H ⇒ Chromatic number is 4.

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Example Scheduling Exams Have eight exams to schedule: French, Math, History, Philosophy, English, Italian, Spanish, Chemistry Some students are taking two or more classes. Only two air-conditioned rooms.

Conflict Table X if overlap M H P E I S C French   X Mathematics History Philosophy English Italian Spanish Chemistry

Conflict Graph classes correspond to vertices edges join conflicted vertices look for vertex coloring any two vertices joined by an edge have different colors colors are different exam times

Conflict Graph

Example Block exam conflicts for fall term (excluding language exams).

Block Exams Revisited Initial graph: B1081 P2001 C1020 P1051 C1040

Clean up and identify complete graphs (several with n = 4): Block Exams Revisited Clean up and identify complete graphs (several with n = 4): M1062 C1020 M1044 B1081 P2001 M1061 M1045 M1046 M1021 M1026 C1040 P1051

Finish the coloring using only the initial 4 colors: Block Exams Revisited Finish the coloring using only the initial 4 colors: M1062 C1020 B1081 M1044 P2001 M1061 M1045 M1046 M1021 M1026 C1040 P1051