The 2nd-order conditional 3-coloring of claw-free graphs

A 2nd-order conditional k-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex of degree at least 2 in G will be adjacent to vertices with at least 2 different colors. The smallest number k for which a graph G can have a 2nd-order conditional k-coloring is the 2nd-order conditional chromatic number, denoted by χd(G). In this paper, we investigate the 2nd-order conditional 3-colorings of claw-free graphs. First, we prove that it is NP-complete to determine if a claw-free graph with maximum degree 3 is 2nd-order conditionally 3-colorable. Second, by forbidding a kind of subgraphs, we find a reasonable subclass of claw-free graphs with maximum degree 3, for which the 2nd-order conditionally 3-colorable problem can be solved in linear time. Third, we give a linear time algorithm to recognize this subclass of graphs, and a linear time algorithm to determine whether it is 2nd-order conditionally 3-colorable. We also give a linear time algorithm to color the graphs in the subclass by 3 colors.