(d,1)‐total labeling of graphs with a given maximum average degree

Abstract

AbstractThe (d,1)‐total number $\lambda _{d}^{T}(G)$ of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G so that no two adjacent vertices have the same color, no two incident edges have the same color, and the distance between the color of a vertex and its incident edges is at least d. In this paper, we prove that $\lambda_{d}^{T}(G) \leq \Delta (G) + 2d - 2$ for connected graphs with a given maximum average degree. © 2005 Wiley Periodicals, Inc. J Graph Theory