Gyárfás–Lehel complete bipartite monochromatic component cover conjecture

Prove that, for every integer r≥2, every r-edge-colouring of the complete bipartite graph K_{n,m} permits its vertex set to be covered by at most 2r−2 monochromatic connected components.

Background

The paper places its balanced-bipartite minimum-degree theorem in the context of a classical conjecture concerning complete bipartite graphs. The conjecture asks for a universal bound of 2r−2 monochromatic connected components in every r-edge-colouring of K_{n,m}. The authors note that this problem is closely related to Ryser-type covering problems for hypergraphs.

References

Considering bipartite graphs, a classical conjecture of Gyárfás and Lehel asserts that, for $r\ge2$, every $r$-edge-colouring of the complete bipartite graph $K_{n,m}$ can be covered by at most $2r-2$ monochromatic connected components; that is, $r(K{n,m})\le 2r-2$.

— Monochromatic components in dense 2-edge-coloured balanced bipartite graphs  (2608.17300 - Bispo et al., 18 Aug 2026) in Section 1, Introduction