The evolution of unavoidable bi-chromatic patterns and extremal cases of balanceability [article]

Yair Caro, Adriana Hansberg, Amanda Montejano
2022 arXiv   pre-print
We study the color patterns that, for n sufficiently large, are unavoidable in 2-colorings of the edges of a complete graph K_n with respect to min{e(R), e(B)}, where e(R) and e(B) are the numbers of red and, respectively, blue edges. More precisely, we determine how such unavoidable patterns evolve from the case without restriction in the coloring, namely that min{e(R), e(B)}≥ 0 (given by Ramsey's theorem), to the highest possible restriction, namely that |e(R) - e(B)| ≤ 1. We also investigate
more » ... the effect of forbidding certain sub-structures in each color. In particular, we show that, in 2-colorings whose graphs induced by each of the colors are both free from an induced matching on r edges, the appearance of the unavoidable patterns is already granted with a much weaker restriction on min{e(R), e(B)}. We finish analyzing the consequences of these results to the balancing number bal(n,G) of a graph G (i.e. the minimum k such that every 2-edge coloring of K_n with min{e(R), e(B)} > k contains a copy of G with half the edges in each color), and show that, for every ε > 0, there are graphs G with bal(n,G) ≥ c n^2-ε, which is the highest order of magnitude that is possible to achieve, as well as graphs where bal(n,G) ≤ c(G), where c(G) is a constant that depends only G. We characterize the latter ones.
arXiv:2204.04269v1 fatcat:vhin3gldsvb7zcbjobgsesye74