radio k-coloring problem is to determine the minimum span of a radio k-coloring of G, denoted by rc k (G). The span of a radio k-coloring f is the number max u∈V(G) f(u). The radio k-chromatic number rck(G) of G is min such that |f(u)−f(v)|⩾k+1−d(u,v) holds for each pair of distinct vertices u and v of G, where diam(G) is the diameter of G and d(u,v) is the distance between u and v in G. The maximum color assigned by f is called its span, denoted by rck(f). For a simple connected graph G and a positive integer k⩽diam(G), a radio k-coloring is an assignment f of positive integers (colors) to the vertices of G such that for every pair of distinct vertices u and v of G, the difference between their colors is at least 1+k−d(u,v). Radio k-coloring of graphs is one of the variations of frequency assignment problem. In this paper, we prove that the Cartesian product of trees with paths are inh-colorable. For an inh-colorable graph G the lower inh-span or simply inh-span of G, denoted by is defined as span is an inh-coloring of G}. A graph G is inh-colorable if there exists an inh-coloring of it. An irreducible no-hole coloring of a graph G, in short inh-coloring of G, is an L(2, 1)-coloring of G which is both irreducible and no-hole. An L(2, 1)-coloring is said to be irreducible if the color of none of the vertices in the graph can be decreased and yield another L(2, 1)-coloring of the same graph. A no-hole coloring is defined to be an L(2, 1)-coloring with no hole in it. If f is an L(2, 1)-coloring of a graph G with span k then an integer l is a hole in f if and there is no vertex v in G such that f(v) = l. The span of an L(2, 1)-coloring f of G, denoted by span(f), is max The span of G, denoted by is the minimum span of all possible L(2, 1)-colorings of G. An L(2, 1)-coloring of a graph G is a mapping such that for all edges uv of G, and if u and v are at distance two in G.
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