2024/03/19 by Fernández, Camila, Pavez-Signé, Matías, Stein, Maya
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2403.12587
We study two variations of the Gyarfas--Lehel conjecture on the minimum number of monochromatic components needed to cover an edge-coloured complete bipartite graph. Specifically, we show the following. - For p>> (log n/n)1/2, w.h.p.~every 2-colouring of the random bipartite graph G~ G(n,n,p) admits a cover of all but O(1/p) vertices of G using at most three vertex-disjoint monochromatic components. - For every 2-colouring of a bipartite graph G with parts of size n and minimum degree (13/16+o(1))n, the vertices of G can be covered using at most three vertex-disjoint monochromatic components.