2012/12/31 by Chen, G., Fujita, S., Gyarfas, A. +2
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1212.6861
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 2r-1 (and this is best possible).