Bounded diameter variations of Ryser's conjecture

TL;DR

This paper investigates diameter bounds in Ryser's conjecture, proving new diameter constraints for 2-colorings and exploring the minimum number of monochromatic components needed for coverage.

Contribution

It improves existing diameter bounds for monochromatic components in 2-colorings of graphs with independence number 2 and proposes bounds on the number of components needed for coverage with fixed diameters.

Findings

In every 2-coloring of a graph with independence number 2, the vertex set can be covered by two monochromatic subgraphs of diameter at most 4.

The diameter bound of 4 can be improved to 3 for certain graphs, including odd antiholes.

For fixed diameter d, the minimum number of monochromatic components needed is determined for specific cases, such as r=2,3 and d=2,4.

Abstract

In this paper we study bounded diameter variations of the following form of Ryser's conjecture. For every graph $G=(V,E)$ with independence number $\alpha(G)=\alpha$ and integer $r\geq 2$, in every $r$-edge coloring of $G$ there is a cover of $V(G)$ by the vertices of $(r-1)\alpha$ monochromatic connected components. Mili\'{c}evi\'{c} initiated the question whether the diameters of the covering components can be bounded. For any graph $G$ with $\alpha(G)=2$ we show that in every 2-coloring of the edges, $V(G)$ can be covered by the vertices of two monochromatic subgraphs of diameter at most 4. This improves a result of DeBiasio et al., which in turn improved a result of Mili\'{c}evi\'{c}. It remains open whether diameter $4$ can be strengthened to diameter $3$, we could do this only for certain graphs, including odd antiholes. We propose also a somewhat orthogonal aspect of the problem. Suppose that we fix the diameter $d$ of the monochromatic components, how many do we need to cover the vertex set? For $d=2,2\le r \le 3$, the exact answer is $r\alpha$ and for $d=4,r=2$, we prove the upper bound $\lfloor 3\alpha/2\rfloor$.