Bounded diameter variations of Ryser's conjecture

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 $r\ge 2$, in every $r$-edge coloring of $G$ there is a cover of $V(G)$ by the vertices of $(r-1)\alpha$ monochromatic connected components. Milićević 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 Milićević. 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 $\lfloor 3\alpha /2\rfloor$.