Around a biclique cover conjecture
Combinatorics
2013-01-01 v1
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).
Cite
@article{arxiv.1212.6861,
title = {Around a biclique cover conjecture},
author = {G. Chen and S. Fujita and A. Gyarfas and J. Lehel and A. Toth},
journal= {arXiv preprint arXiv:1212.6861},
year = {2013}
}