Around a biclique cover conjecture

TL;DR

This paper investigates a longstanding conjecture about covering vertices in edge-colored bicliques with monochromatic connected components, proving it for fewer than six colors and establishing bounds on biclique width.

Contribution

It reduces the conjecture to design-like problems and proves it for r<6, also providing bounds on biclique width in r-colorings.

Findings

Proved the conjecture for r<6.

Established that the biclique width in any spanning r-coloring is at most 2^{r-1}.

Demonstrated the bound is tight (best possible).

Abstract

We address an old (1977) conjecture of a subset of the authors (a variant of Ryser's conjecture): in every r-coloring of the edges of a biclique [A,B] (complete bipartite graph), the vertex set can be covered by the vertices of at most 2r-2 monochromatic connected components. We reduce this conjecture to design-like conjectures, where the monochromatic components of the color classes are bicliques [X,Y] with nonempty blocks X and Y. We prove this conjecture for r<6. We show that the width (the number of bicliques) in every color class of any spanning r-coloring is at most 2^{r-1} (and this is best possible).