The Study on the Upper-bound of Labeling Number for Chordal and Permutation Graphs

코달 및 순열 그래프의 레이블링 번호 상한에 대한 연구

Given a graph G=(V,E), Ld(2,1)-labeling of G is a function f : V(G)$\longrightarrow$[0,$\infty$) such that, if v1,v2$\in$V are adjacent, $\mid$ f(x)-f(y) $\mid$$\geq$2d, and, if the distance between and is two, $\mid$ f(x)-f(y) $\mid$$\geq$d, where dG(,v2) is shortest distance between v1 and in G. The L(2,1)-labeling number (G) is the smallest number m such that G has an L(2,1)-labeling f with maximum m of f(v) for v$\in$V. This problem has been studied by Griggs, Yeh and Sakai for the various classes of graphs. In this paper, we discuss the upper-bound of ${\lambda}$ (G) for a chordal graph G and that of ${\lambda}$(G') for a permutation graph G'.