Chromatic capacity and graph operations

The chromatic capacity cap(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 cap(G) f((G)) for almost every graph G, where denotes the chromatic number. We show that for any positive integers n and k with k n/2 there exists a graph G with (G) = n and cap(G) = n - k, extending a result of Greene. We obtain bounds on cap(Kr n) that are tight as r , where Kr n 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 cap(G) + 1 = (G) = p that contains no odd cycles of length less than q. Key words: Chromatic capacity, Emulsive edge coloring, Compatible vertex coloring, Split coloring, Lexicographic product

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Jack Huizenga
• Keywords: Chromatic Capacity, Chromatic Capacity Cap, DM 2008, Positive Integers