On list extensions of the majority edge colourings

TL;DR

This paper studies list-based extensions of majority edge colourings in graphs, providing bounds on minimum degree for existence of such colourings and addressing conjectures for specific cases.

Contribution

It proves that graphs with sufficiently high minimum degree admit list majority edge colourings, nearly matching non-list results and solving a conjecture for the basic case.

Findings

Graphs with minimum degree ≥ 2k^2-2k admit 1/k-majority list colourings for lists of size k+1

Results extend to diversified majority tolerances with certain list and degree conditions

Bounds are strengthened for fixed majority tolerances and list sizes in a general setting

Abstract

We investigate possible list extensions of generalised majority edge colourings of graphs and provide several results concerning these. Given a graph $G=(V,E)$, a list assignment $L:E\to 2^C$ and some level of majority tolerance $\alpha\in(0,1)$, an $\alpha$-majority $L$-colouring of $G$ is a colouring $\omega:E\to C$ from the given lists such that for every $v\in V$ and each $c\in C$, the number of edges coloured $c$ which are incident with $v$ does not exceed $\alpha\cdot d(v)$. We present a simple argument implying that for every integer $k\geq 2$, each graph with minimum degree $\delta\geq 2k^2-2k$ admits a $1/k$-majority $L$-colouring from any assignment of lists of size $k+1$. This almost matches the best result in a non-list setting and solves a conjecture posed for the basic majority edge colourings, i.e. for $k=2$, from lists. We further discuss restrictions which permit obtaining corresponding results in a more general setting, i.e. for diversified $\alpha=\alpha(c)$ majority tolerances for distinct colours $c\in C$. Consider a list assignment $L:E\to 2^C$ with $\sum_{c\in L(e)}\alpha(c)\geq 1+\varepsilon$ for each edge $e$, and suppose that $\alpha(c)\geq a$ for every $c$ or $|L(e)|\leq\ell$ for all edges $e$, where $a\in(0,1)$, $\varepsilon>0$, $\ell\in\mathbb{N}$ are any given constants. Then we in particular show that there exists an $\alpha$-majority $L$-colouring of $G$ from any such list assignment, provided that $\delta(G)=\Omega(a^{-1}\varepsilon^{-2}\ln(a\varepsilon)^{-1})$ or $\delta=\Omega(\ell^2\varepsilon^{-2})$, respectively. We also strengthen these bounds within a setting where each edge is associated to a list of colours with a fixed vector of majority tolerances, applicable also in a general non-list case.