A bounded diameter strengthening of Kőnig's Theorem
Louis DeBiasio, António Girão, Penny Haxell, Maya Stein
TL;DR
This paper strengthens König's theorem in the setting of 2-color edge colorings by showing that, for any graph $G$, the vertex set can be covered by at most $\alpha(G)$ monochromatic subgraphs of bounded diameter, where the diameter bound is $f(\alpha)=8\alpha^2+12\alpha+6$. The proof uses induction on the independence number, introducing a detailed combinatorial construction involving maximal independent sets, vertex labeling, and alternating paths to iteratively reduce the problem to smaller graphs while preserving a diameter bound. The result confirms a bounded-diameter strengthening of König's theorem and situates it within broader questions about multi-color extensions and Ryser-type conjectures, highlighting several open problems and potential directions for refinement. The work advances the understanding of how diameter constraints interact with colourings and covers in graphs, with implications for related conjectures and diameter-restricted decompositions.
Abstract
K\H onig's theorem says that the vertex cover number of every bipartite graph is at most its matching number (in fact they are equal since, trivially, the matching number is at most the vertex cover number). An equivalent formulation of K\H onig's theorem is that in every $2$-colouring of the edges of a graph $G$, the number of monochromatic components needed to cover the vertex set of $G$ is at most the independence number of $G$. We prove the following strengthening of K\H onig's theorem: In every $2$-colouring of the edges of a graph $G$, the number of monochromatic subgraphs of bounded diameter needed to cover the vertex set of $G$ is at most the independence number of $G$.