Chromatic capacity and graph operations

The chromatic capacityχ_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 χ_cap (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 χ_cap (G) = n - k, extending a result of Greene. We obtain bounds on χ_cap (K_n^r) that are tight as r → ∞, where K_n^r 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 χ_cap (G) + 1 = χ (G) = p that contains no odd cycles of length less than q.