A bounded diameter strengthening of K\H{o}nig's Theorem

TL;DR

This paper strengthens Kőnig's theorem by showing that in any 2-edge coloring of a graph, the vertex set can be covered by a bounded number of monochromatic subgraphs with limited diameter, not just components.

Contribution

It introduces a new bound on the number of monochromatic subgraphs of bounded diameter needed to cover a graph, extending Kőnig's theorem to a diameter-restricted setting.

Findings

Bounded diameter monochromatic coverings are at most the independence number.

The result generalizes classical Kőnig's theorem.

Provides a new perspective on graph colorings and coverings.

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 $G$.