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arXiv · 1507.07573

Asymptotically optimal neighbor sum distinguishing total colorings of graphs

Sarah Loeb·Jakub Przybyło·Yunfang Tang
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Abstract

Given a proper total $k$-coloring $c:V(G)\cup E(G)\to\{1,2,\ldots,k\}$ of a graph $G$, we define the value of a vertex $v$ to be $c(v) + \sum_{uv \in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose values form a proper coloring is the neighbor sum distinguishing total chromatic number of $G$, $\chi"_{\Sigma}(G)$. Pil\'sniak and Wo\'zniak (2013) conjectured that $\chi"_{\Sigma}(G)\leq \Delta(G)+3$ for any simple graph with maximum degree $\Delta(G)$. In this paper, we prove this bound to be asymptotically correct by showing that $\chi"_{\Sigma}(G)\leq \Delta(G)(1+o(1))$. The main idea of our argument relies on Przyby{\l}o's proof (2014) regarding neighbor sum distinguishing edge-colorings.

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Sarah Loeb, Jakub Przybyło, Yunfang Tang. 2015-07-27. Asymptotically optimal neighbor sum distinguishing total colorings of graphs. https://arxiv.org/abs/1507.07573

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