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 ${1,2,\dots, k}$. 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 $\chi_t(G)$, 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 $\chi_t(G)$ is $2$-approximate and runs in $O(tn)$ time steps (where $n$ is the vertex number of $G$). In particular, we prove that for $t\in {1,2,3,4,5,6}$, the greedy algorithm produces an optimal $t$-relaxed coloring of a complete multi-partite graph. While, for $t\ge 7$, examples are given to illustrate that the greedy strategy does not always construct an optimal $t$-relaxed coloring.