Asymptotically optimal neighbor sum distinguishing total colorings of graphs

TL;DR

This paper proves that the neighbor sum distinguishing total chromatic number of a graph is asymptotically at most its maximum degree, confirming a long-standing conjecture within a factor approaching one.

Contribution

It establishes an asymptotic upper bound for the neighbor sum distinguishing total chromatic number, improving understanding of graph colorings related to vertex sums.

Findings

Proves $oxed{ ext{χ}''_{ ext{Σ}}(G) o ext{Δ}(G)}$ asymptotically

Confirms Pilśniak and Woźniak's conjecture in the limit

Builds on Przybyło's proof for edge-colorings

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.