On List Extensions of the Majority Edge Colourings

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.