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.
@article{10_37236_10563,
author = {Huajing Lu and Xuding Zhu},
title = {Dense {Eulerian} graphs are \((1, 3)\)-choosable},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {2},
doi = {10.37236/10563},
zbl = {1492.05083},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10563/}
}
TY - JOUR
AU - Huajing Lu
AU - Xuding Zhu
TI - Dense Eulerian graphs are \((1, 3)\)-choosable
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/10563/
DO - 10.37236/10563
ID - 10_37236_10563
ER -
%0 Journal Article
%A Huajing Lu
%A Xuding Zhu
%T Dense Eulerian graphs are \((1, 3)\)-choosable
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/10563/
%R 10.37236/10563
%F 10_37236_10563
Huajing Lu; Xuding Zhu. Dense Eulerian graphs are \((1, 3)\)-choosable. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10563
