On t-relaxed coloring of complete multi-partite graphs

Abstract

Let G be a graph and t a nonnegative integer. Suppose f is a mapping from the vertex set of G to {1, 2, . . . , k}. If, for any vertex u of G, the number of neighbors v of u with f (v) = f (u) is less than or equal to t, then f is called a t-relaxed k-coloring of G. And G is said to be (k, t)-colorable. The t-relaxed chromatic number of G, denote by χ t (G), is defined as the minimum integer k such that G is (k, t)-colorable. A set S of vertices in G is t-sparse if S induces a graph with a maximum degree of at most t. Thus G is (k, t)-colorable if and only if the vertex set of G can be partitioned into k t-sparse sets. It was proved by Belmonte, Lampis and Mitsou (2017) that the problem of deciding if a complete multi-partite graph is (k, t)-colorable is NP-complete. In this paper, we first give tight lower and up bounds for the t-relaxed chromatic number of complete multi-partite graphs. And then we design an algorithm to compute maximum t-sparse sets of complete multi-partite graphs running in O((t + 1) 2 ) time. Applying this algorithm, we show that the greedy algorithm for χ t (G) is 2-approximate and runs in O(tn) time steps (where n is the vertex number of G). In particular, we prove that for t ∈ {1, 2, 3, 4, 5, 6}, the greedy algorithm produces an optimal t-relaxed coloring of a complete multi-partite graph. While, for t ≥ 7, examples are given to illustrate that the greedy strategy does not always construct an optimal t-relaxed coloring.