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

TL;DR

This paper proves that large Eulerian graphs with high minimum degree are total weight (1,3)-choosable, and establishes conditions under which graphs are total weight (1,4)-choosable, advancing understanding of graph weightings.

Contribution

It introduces new results on total weight (k,k')-choosability for Eulerian graphs and graphs with high minimum degree, including specific bounds and conditions for choosability.

Findings

Eulerian graphs of large order are total weight (1,3)-choosable under certain degree conditions.

Graphs with minimum degree at least 0.999 times the number of vertices are total weight (1,4)-choosable.

Decomposition into complete graphs of odd order implies total weight (1,3)-choosability.

Abstract

A graph $G$ is total weight $(k,k')$-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 $k'$ 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.