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 χcap (Knr) that are tight as r → ∞, where Knr 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.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics