Channel assignment problem and relaxed 2-distant coloring of graphs

Abstract

Let G be a simple graph. Suppose f is a mapping from V(G) to nonnegative integers. If, for any two adjacent vertices u and v of G, |f(u)−f(v)|≥2, then f is called a 2-distant coloring of G. In this paper, we introduce a relaxation of 2-distant coloring of a graph. Let t be a nonnegative integer. Suppose f is a mapping from V(G) to nonnegative integers. If adjacent vertices receive different integers and for each vertex u of G, the number of neighbors v of u with |f(v)−f(u)|=1 is at most t, then f is called a t-relaxed 2-distant coloring of G. If t=0 then f is just a 2-distant coloring of G. The span of f, denote by sp(f), is the difference between the maximum and minimum integers used by f. The minimum span of a t-relaxed 2-distant coloring of G, is called t-relaxed 2-distant coloring span of G, denoted by sp2t(G). Suppose G is a graph. Let γ:V(G)→Z+ be a function defined on V(G). A γ-relaxed 2-distant coloring of G with span k is a mapping f from V(G) to {0,1,…,k} such that f(u)≠f(v) for any two adjacent vertices u and v and each vertex u of G has at most γ(u) neighbors v with |f(v)−f(u)|=1.In this paper, we prove that, for any two positive integers t and k with k≥2, deciding if sp2t(G)≤k for a graph G is NP-complete, except the case when t=1 and k=2 which is polynomial-time solvable. Let t be any nonnegative integer. It is easy to see that sp2t(G)≤6 for any planar graph G and sp2t(G)≤4 for any outerplanar graph G. We prove that, for any two positive integers t and k with k∈{2,3,4,5}, deciding if sp2t(G)≤k for a planar graph G is NP-complete, except the case when t=1 and k=2 which is polynomial-time solvable. We present examples to illustrate that there is no constant integer t such that every planar graph has a t-relaxed 2-distant coloring of span 5 and there is no constant integer t such that every outerplanar graph has a t-relaxed 2-distant coloring of span 3. We introduce pseudo ear decomposition and simple pseudo ear decomposition of a graph and show that a graph is outerplanar if and only if it admits a simple pseudo ear decomposition. Using this result, we show that each outerplanar graph has a γ-relaxed 2-distant coloring with span 3, where γ(v)=⌈d(v)2⌉ for each vertex v of G and the function γ is sharp in some sense, and that every triangle-free outerplanar graph has a 1-relaxed 2-distant coloring with span 3.