Chromatic capacity and graph operations

The chromatic capacity χcap(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 χcap(K 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 χcap(G) + 1 = χ(G) = p that contains no odd cycles of length less than q.