On generalised majority edge-colourings of graphs

TL;DR

This paper investigates generalized majority edge-colourings in graphs, proposing a conjecture on minimum degree conditions for such colourings, supporting it with improved bounds, and proving it for small cases and bipartite graphs.

Contribution

It introduces a conjecture on majority edge-colourings, provides improved bounds supporting the conjecture, and proves the conjecture for small values of k and bipartite graphs.

Findings

Proved the conjecture for k ≤ 4.

Improved bounds from k^3 log k to approximately 2k^2 and further to (7/4+o(1))k^2.

Supported the conjecture with bounds close to the conjectured minimum degree.

Abstract

A $\frac{1}{k}$-majority $l$-edge-colouring of a graph $G$ is a colouring of its edges with $l$ colours such that for every colour $i$ and each vertex $v$ of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. We conjecture that for every integer $k\geq 2$, each graph with minimum degree $\delta\geq k^2$ is $\frac{1}{k}$-majority $(k+1)$-edge-colourable and observe that such result would be best possible. This was already known to hold for $k=2$. We support the conjecture by proving it with $2k^2$ instead of $k^2$, which confirms the right order of magnitude of the conjectured optimal lower bound for $\delta$. We at the same time improve the previously known bound of order $k^3\log k$, based on a straightforward probabilistic approach. As this technique seems not applicable towards any further improvement, we use a more direct non-random approach. We also strengthen our result, in particular substituting $2k^2$ by $(\frac{7}{4}+o(1))k^2$. Finally, we provide the proof of the conjecture itself for $k\leq 4$ and completely solve an analogous problem for the family of bipartite graphs.