On $L(1,2)$-Edge-Labelings of Some Special Classes of Graphs

DOI：10.3770/j.issn:2095-2651.2014.04.003

 作者 单位 贺丹 东南大学数学系, 江苏 南京 210096 林文松 东南大学数学系, 江苏 南京 210096

设$j,k$为两个正整数,图$G$的$m$-$L(j,k)$-边标号是图的边集到非负整数集$\{0,1,2,\ldots,m\}$的一个映射,使得相邻的边对应的整数相差至少为$j$,距离为2的边所对应的整数相差至少为$k$.图$G$的所有$m$-$L(j,k)$-边标号中最小的$m$值称为图$G$的$L(j,k)$-边标号数,记为$l'j,k(G)$.本文主要研究了路、圈、完全图、完全多部图、无穷正则树及轮图等一些特殊图类的$L(1,2)$-边标号.

For a graph $G$ and two positive integers $j$ and $k$, an $m$-$L(j,k)$-edge-labeling of $G$ is an assignment on the edges to the set $\{0,\ldots,m\}$, such that adjacent edges receive labels differing by at least $j$, and edges which are distance two apart receive labels differing by at least $k$. The $\lambda^{\prime}_{j,k}$-number of $G$ is the minimum $m$ of an $m$-$L(j,k)$-edge-labeling admitted by $G$. In this article, we study the $L(1,2)$-edge-labeling for paths, cycles, complete graphs, complete multipartite graphs, infinite $\Delta$-regular trees and wheels.