Distant total sum distinguishing index of graphs

Abstract

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. We say vertices $u,v\in V$ are sum distinguished if $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. By $\chi"_{\Sigma,r}(G)$ we denote the least integer $k$ admitting such a colouring $c$ for which every $u,v\in V$, $u\neq v$, at distance at most $r$ from each other are sum distinguished in $G$. For every positive integer $r$ an infinite family of examples is known with $\chi"_{\Sigma,r}(G)=\Omega(\Delta^{r-1})$. In this paper we prove that $\chi"_{\Sigma,r}(G)\leq (2+o(1))\Delta^{r-1}$ for every integer $r\geq 3$ and each graph $G$, while $\chi"_{\Sigma,2}(G)\leq (18+o(1))\Delta$.