(d,1)-total labelling of sparse graphs

Louis Esperet (1) , Mickael Montassier (1) , André Raspaud (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The $(d,1)$-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 the color of any incident edge is at least $d$. This notion was introduced by Havet and Yu in \cite{HY02}. In this paper, we study the $(d,1)$-total number of sparse graphs and prove that for any $0 < \varepsilon < \frac{1}{2}$, and any positive integer $d$, there exists a constant $C_{d,\varepsilon}$ such that for any $\varepsilon \Delta$-sparse graph $G$ with maximum degree $\Delta$, we have $\lambda_d^T(G)\leq \Delta + C_{d,\varepsilon}$.

Domaines: Mathématique discrète [cs.DM]

Louis Esperet, Mickael Montassier, André Raspaud. (d,1)-total labelling of sparse graphs. 2006. ⟨hal-00401711⟩