Bounded diameter variations of Ryser's conjecture

TL;DR

The paper investigates bounded-diameter variants of Ryser's conjecture for edge colorings. Focusing on graphs with independence number $α$ and two colors, it proves that when $α(G)=2$, every 2-coloring yields a pair of monochromatic components whose diameters are at most $4$ that cover $V(G)$, improving previous bounds. It also develops a fixed-diameter framework, giving exact values for diameter $2$ and $d=4$ in certain $r$ and $α$ regimes, with a core structural toolkit based on stars, double stars, and a refined classification of 2-colored complete graphs, including the special families ${\u0008cal{H}}_1$, ${\u0008cal{H}}_2$ and odd antihole considerations. The work provides two distinct proofs for the $rac{3}{2}α$-type bound when the diameter is $4$ and outlines key open questions about strengthening diameter bounds and about the function $g(r,d,α)$ that governs fixed-diameter coverings. Overall, it advances understanding of how diameter constraints interact with classical Ryser-type coverings in sparse colorings and clarifies both achievable bounds and limitations.

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 $α(G)=α$ and integer $r\geq 2$, in every $r$-edge coloring of $G$ there is a cover of $V(G)$ by the vertices of $(r-1)α$ monochromatic connected components. Milićević initiated the question whether the diameters of the covering components can be bounded. For any graph $G$ with $α(G)=2$ 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 $rα$ and for $d=4,r=2$, we prove the upper bound $\lfloor 3α/2\rfloor$.

Figures (2)

• Figure 1: The special 2-colored $K_4$ denoted by $P_4^2$
• Figure 2: The "skeleton" of the graphs in ${\cal{H}}_1$

Theorems & Definitions (16)

• Conjecture 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 2.1
• Theorem 2.2
• Remark 2.4
• Theorem 2.5
• Proposition 3.1
• Proposition 3.2
• proof