2-tone coloring of cactus graphs

Abstract

A 2-tone coloring of a graph assigns two distinct colors to each vertex with the restriction that adjacent vertices have no common colors, and vertices at distance two have at most one common color. The 2-tone chromatic number of a graph is the minimum number of colors in any 2-tone coloring. A cactus graph has every block a cycle or edge. We determine the 2-tone chromatic number of all cactus graphs with maximum degree Δ≠6. When Δ=6, we determine the 2-tone chromatic number of all cactus graphs that are triangle-free and those with circumference at most 8.