Chromatic capacity and graph operations |
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Authors: | Jack Huizenga |
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Institution: | Department of Mathematics, Harvard University, Cambridge, MA 02138, USA |
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Abstract: | The chromatic capacityχcap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f:N→N such that χcap(G)?f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show that for any positive integers n and k with k?n/2 there exists a graph G with χ(G)=n and χcap(G)=n-k, extending a result of Greene. We obtain bounds on that are tight as r→∞, where is the complete n-partite graph with r vertices in each part. Finally, for any positive integers p and q we construct a graph G with χcap(G)+1=χ(G)=p that contains no odd cycles of length less than q. |
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Keywords: | Chromatic capacity Emulsive edge coloring Compatible vertex coloring Split coloring Lexicographic product |
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