On irreducible no-hole L(2, 1)-coloring of subdivision of graphs

Abstract

An L(2, 1)-coloring (or labeling) of a graph G is a mapping $$f:V(G) \rightarrow \mathbb {Z}^{+}\bigcup \{0\}$$f:V(G)źZ+ź{0} such that $$|f(u)-f(v)|\ge 2$$|f(u)-f(v)|ź2 for all edges uv of G, and $$|f(u)-f(v)|\ge 1$$|f(u)-f(v)|ź1 if u and v are at distance two in G. The span of anL(2, 1)-coloringf, denoted by span f, is the largest integer assigned by f to some vertex of the graph. The span of a graphG, denoted by $$\lambda (G)$$ź(G), is min {span $$f: f\text {is an }L(2,1)\text {-coloring of } G\}$$f:fis anL(2,1)-coloring ofG}. 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 $$l\in (0,k)$$lź(0,k) and there is no vertex v in G such that $$f(v)=l$$f(v)=l. A no-hole coloring is defined to be an L(2, 1)-coloring with span k which uses all the colors from $$\{0,1,\ldots ,k\}$${0,1,ź,k}, for some integer k not necessarily the span of the graph. An L(2, 1)-coloring is said to be irreducible if colors of no 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, also called inh-coloring of G, is an L(2, 1)-coloring of G which is both irreducible and no-hole. The lower inh-span or simply inh-span of a graph G, denoted by $$\lambda _{inh}(G)$$źinh(G), is defined as $$\lambda _{inh}(G)=\min ~\{$$źinh(G)=min{span f : f is an inh-coloring of G}. Given a graph G and a function h from E(G) to $$\mathbb {N}$$N, the h-subdivision of G, denoted by $$G_{(h)}$$G(h), is the graph obtained from G by replacing each edge uv in G with a path of length h(uv). In this paper we show that $$G_{(h)}$$G(h) is inh-colorable for $$h(e)\ge 2$$h(e)ź2, $$e\in E(G)$$eźE(G), except the case $$\Delta =3$$Δ=3 and $$h(e)=2$$h(e)=2 for at least one edge but not for all. Moreover we find the exact value of $$\lambda _{inh}(G_{(h)})$$źinh(G(h)) in several cases and give upper bounds of the same in the remaining.