Vertex covering with monochromatic pieces of few colours
Marlo Eugster, Frank Mousset

TL;DR
This paper generalizes a vertex cover problem in edge-coloured complete graphs to multiple colours, establishing bounds based on Kneser hypergraph chromatic numbers and extending results to arbitrary graphs with fixed independence number.
Contribution
It introduces a multi-colour vertex cover problem, linking the minimum number of monochromatic paths to Kneser hypergraph chromatic numbers, and extends the analysis to general graphs with fixed independence number.
Findings
est asymptotic bounds for _{r,s}(K_n) as n^{1/}
emonstrates tightness of bounds with specific graph constructions
xtends results to covering with monochromatic cycles and regular graphs
Abstract
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every -colouring of the edges of , there is a vertex cover by monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers , what is the smallest number such that in every colouring of the edges of with colours, there exists a vertex cover of by monochromatic paths using altogether at most different colours? For fixed integers and as , we prove that , where is the chromatic number of the Kneser gr aph . More generally, if one replaces by an arbitrary -vertex graph with fixed independence…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Advanced Topology and Set Theory
