Vertex Colouring Edge Weightings: a Logarithmic Upper Bound on Weight-Choosability

Abstract

A graph $G$ is said to be $(k,m)$-choosable if for any assignment of $k$-element lists $L_v \subset \mathbb{R}$ to the vertices $v \in V(G)$ and any assignment of $m$-element lists $L_e \subset \mathbb{R}$ to the edges $e \in E(G)$ there exists a total weighting $w: V(G) \cup E(G) \rightarrow \mathbb{R}$ of $G$ such that $w(v) \in L_v$ for any vertex $v \in V(G)$ and $w(e) \in L_e$ for any edge $e \in E(G)$ and furthermore, such that for any pair of adjacent vertices $u,v$, we have $w(u)+ \sum_{e \in E(u)}w(e) \neq w(v)+ \sum_{e \in E(v)}w(e)$, where $E(u)$ and $E(v)$ denote the edges incident to $u$ and $v$ respectively. In this paper we give an algorithmic proof showing that any graph $G$ without isolated edges is $(1, 2 \lceil \log_2(\Delta(G)) \rceil+1)$-choosable, where $\Delta(G)$ denotes the maximum degree in $G$.

An analysis between different algorithms for the graph vertex coloring problem

“…Similar approaches have been used to solve other problems in graph theory [25]. Detailed reviews of the specifics of the graph vertex coloring problem are discussed in [26], [27].…”

The 1-2-3 Conjecture holds for graphs with large enough minimum degree

“…its extension to a natural list setting [8,42,48], with interesting applications [8,47] of algebraic approach exploiting Alon's Combinatorial Nullstellensatz. Intriguingly, though it is believed that 3-element lists of weights should suffice, see [8], no finite upper bound is known thus far in this more demanding setting; see [33] for a result implying that lists of length O(log ∆) are enough.…”