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. 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. 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. A no-hole coloring is defined to be an L(2, 1)-coloring with no hole in it. 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. 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. A graph G is inh-colorable if there exists an inh-coloring of it. 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}. In this paper, we prove that the Cartesian product of trees with paths are inh-colorable.