Bounded diameter monochromatic component covers

Ryser conjectured that every ‐edge‐coloured complete graph can be covered by monochromatic trees. Motivated by a question of Austin in analysis, Milićević predicted something stronger — that every ‐edge‐coloured complete graph can be covered by monochromatic trees of bounded diameter . Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Milićević's Conjecture, most notably that it is true for . We also obtain several new cases of a generalization of Milićević's Conjecture to non‐complete graphs due to DeBiasio–Kamel–McCourt–Sheats.