Open Distance Pattern Coloring of a Graph

Abstract

Given a connected $(p,q)$-graph $G =(V,E)$ of diameter $d(G),~ \emptyset\neq M \subseteq V(G)$ and a nonempty set $X = \{1,2,3,...,d(G)\}$ of colors of cardinality $d(G),$ let $f_M^o$ be an assignment of subsets of $X$ to the vertices of $G,$ such that $f_M^o(u) = \{d(u,v):v \in M, u \neq v\}$ where $d(u,v)$ is the usual distance between $u$ and $v.$ Given such a function $f_M^o$ for all vertices in $G,$ an induced edge function $f ^{\bigoplus}_M$ of an edge $uv \in E(G),~ f_M^ {\bigoplus}(uv) = f_M^o(u)\bigoplus f_M^o(v).$ We call $f_M^o$ an $M$-open distance pattern coloring of $G,$ if no two adjacent vertices have same $f_M^o$ and if such an $M$ exists for a graph $G,$ then $G$ is called an open distance pattern colorable graph; the minimum cardinality of such an $M$ if it exists, is the open distance pattern coloring number of $G,$ denoted by $\eta_M(G).$ Also, if $|f_M^ {\bigoplus}(e)| =k $ for all $e \in E(G),$ we call $G$ a $k$-uniform $M$-open distance pattern colorable graph and if such an $M$ exists for a graph $G,$ then $G$ is called $k$-uniform open distance pattern colorable graph. In this paper we initiate a study on properties of graphs which belongs to this coloring class.