On t-Relaxed 2-Distant Circular Coloring of Graphs

Abstract

Let k be a positive integer. For any two integers i and j in $$\{0,1,\ldots ,k-1\}$$ , let $$|i-j|_k$$ be the circular distance between i and j, which is defined as $$\min \{|i-j|,k-|i-j|\}$$ . Suppose f is a mapping from V(G) to $$\{0,1,\ldots ,k-1\}$$ . If, for any two adjacent vertices u and v in V(G), $$|f(u)-f(v)|_k\ge 2$$ , then f is called a $$\frac{k}{2}$$ -coloring of G. In this paper, we study the relaxation of $$\frac{k}{2}$$ -coloring. If adjacent vertices receive different integers, and for each vertex u of G, the number of neighbors v of u with $$|f(u)-f(v)|_k=1$$ is at most t, then f is called a t-relaxed 2-distant circular k-coloring, or simply a $$(\frac{k}{2},t)^*$$ -coloring of G. If G has a $$(\frac{k}{2},t)^*$$ -coloring, then G is called $$(\frac{k}{2},t)^*$$ -colorable. We prove that, for any two fixed integers k and t with $$k\ge 2$$ and $$t\ge 1$$ , the problem of deciding whether a graph G is $$(\frac{k}{2},t)^*$$ -colorable is NP-complete except the case $$k=2$$ and the case $$k=3$$ and $$t\le 3$$ , which are polynomially solvable. For any outerplanar graph G, it is easy to see that G is $$(\frac{6}{2},0)^*$$ -colorable. We show that all outerplanar graphs are $$(\frac{5}{2},4)^*$$ -colorable, and there is no fixed positive integer t such that all outerplanar graphs are $$(\frac{4}{2},t)^*$$ -colorable.