On list extensions of the majority edge colourings
Paweł Pękała, Jakub Przybyło
TL;DR
The paper studies list-extensions of generalised majority edge colourings, establishing that for every $k\ge 2$ and graphs with minimum degree $δ\ge 2k^2-2k$, there exists a $1/k$-majority edge colouring from lists of size $k+1$. It extends this framework to arbitrary uniform tolerances and to diversified tolerances, providing probabilistic existence results via the Lovász Local Lemma and Chernoff bounds, leveraging vertex-splitting and Galvin's theorem. A unified model with $\varepsilon$-excessive lists and tolerance vectors $\Lambda$ is developed, including a key lemma (The Same Lists) that aids optimization and yields explicit degree bounds such as $δ\ge O(a^{-1}ε^{-2}\ln(aε)^{-1})$, plus specialized bounds for uniform tolerance vectors. Collectively, the results nearly match non-list bounds, broaden the scope to diversified tolerances, and set a foundation for further extensions in both finite and potential infinite graphs.
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 $α\in(0,1)$, an $α$-majority $L$-colouring of $G$ is a colouring $ω:E\to C$ 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 $α\cdot d(v)$. We present a simple argument implying that for every integer $k\geq 2$, each graph with minimum degree $δ\geq 2k^2-2k$ 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 $α=α(c)$ majority tolerances for distinct colours $c\in C$. Consider a list assignment $L:E\to 2^C$ with $\sum_{c\in L(e)}α(c)\geq 1+\varepsilon$ for each edge $e$, and suppose that $α(c)\geq a$ 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 $α$-majority $L$-colouring of $G$ from any such list assignment, provided that $δ(G)=Ω(a^{-1}\varepsilon^{-2}\ln(a\varepsilon)^{-1})$ or $δ=Ω(\ell^2\varepsilon^{-2})$, 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.