Bounded diameter monochromatic component covers

TL;DR

This paper proves the equivalence of two conjectures related to covering complete graphs with monochromatic trees, and confirms the stronger conjecture for the case when r=5, advancing understanding in graph coloring and covering.

Contribution

It establishes the equivalence of Ryser's and Milićević's conjectures and confirms the stronger bounded diameter version for r=5, also extending results to non-complete graphs.

Findings

Proved the equivalence of the two conjectures.

Confirmed Milićević's stronger conjecture for r=5.

Extended results to non-complete graphs.

Abstract

Ryser conjectured that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees. Motivated by a question of Austin in analysis, Mili\'cevi\'c predicted something stronger -- that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Mili\'cevi\'c's Conjecture, most notably that it is true for $r=5$. We also obtain several new cases of a generalization of Mili\'cevi\'c's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats.