A new graph parameter and a construction of larger graph without increasing radio k-chromatic number

Abstract

For a positive integer $k\ge 2$, the radio k-coloring problem is an assignment L of non-negative integers (colors) to the vertices of a finite simple graph G satisfying the condition $|L(u)-L(v)| \ge k+1-d(u,v)$, for any two distinct vertices u, v of G and d(u, v) being distance between u, v. The span of L is the largest integer assigned by L, while 0 is taken as the smallest color. An $rc_k$-coloring on G is a radio k-coloring on G of minimum span which is referred as the radio k-chromatic number of G and denoted by $rc_k(G)$. An integer h, $0<h<rc_k(G)$, is a hole in a $rc_k$-coloring on G if h is not assigned by it. In this paper, we construct a larger graph from a graph of a certain class by using a combinatorial property associated with $(k-1)$ consecutive holes in any $rc_k$-coloring of a graph. Exploiting the same property, we introduce a new graph parameter, referred as $(k-1)$-hole index of G and denoted by $\rho _k(G)$. We also explore several properties of $\rho _k(G)$ including its upper bound and relation with the path covering number of the complement $G^c$.