L(2,1)-labelings of the edge-multiplicity-paths-replacement of a graph

Abstract

An L(2,1)-labeling of a graph $G$ is an assignment of nonnegative integers to $V(G)$ such that the difference between labels of adjacent vertices is at least $2$, and the difference between labels of vertices that are distance two apart is at least 1. The span of an L(2,1)-labeling of a graph $G$ is the difference between the maximum and minimum integers used by it. The minimum span of an L(2,1)-labeling of $G$ is denoted by $\lambda (G)$. This paper focuses on L(2,1)-labelings-number of the edge-multiplicity-paths-replacement $G(rP_{k})$ of a graph $G$. In this paper, we obtain that $r\Delta +1 \le \lambda (G(rP_{5}))\le r\Delta +2$, $\lambda (G(rP_{k}))= r\Delta +1$ for $k\ge 6$; and $\lambda (G(rP_{4}))\le (\Delta +1)r+1$, $\lambda (G(rP_{3}))\le (\Delta +1)r+\Delta$ for any graph $G$ with maximum degree $\Delta$. And the L(2,1)-labelings-numbers of the edge-multiplicity-paths-replacement $G(rP_{k})$ are completely determined for $1\le \Delta \le 2$. And we show that the class of graphs $G(rP_{k})$ with $k\ge 3$ satisfies the conjecture: $\lambda ^{T}_{2}(G)\le \Delta +2$ by Havet and Yu (Technical Report 4650, 2002).