Dense Eulerian graphs are $(1,3)$-choosable

Abstract

A graph $G$ is total weight $(k,k^{\prime })$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and each edge $e$ a set $L(e)$ of $k^{\prime }$ real numbers, there is a proper total $L$-weighting, i.e., a mapping $f:V(G)\cup E(G)\to \mathbb{R}$ such that for each $z\in V(G)\cup E(G)$, $f(z)\in L(z)$, and for each edge $uv$ of $G$, ${\sum }_{e\in E(u)}f(e)+f(u)\ne {\sum }_{e\in E(v)}f(e)+f(v)$. This paper proves that if $G$ decomposes into complete graphs of odd order, then $G$ is total weight $(1,3)$-choosable. As a consequence, every Eulerian graph $G$ of large order and with minimum degree at least $0.91|V(G)|$ is total weight $(1,3)$-choosable. We also prove that any graph $G$ with minimum degree at least $0.999|V(G)|$ is total weight $(1,4)$-choosable.

10 pages. arXiv admin note: text overlap with arXiv:2104.05410