FirstFit coloring process of random bipartite graphs and an online coloring lower bound in the random order model

Source abstract

FirstFit colors each arriving vertex with the smallest positive integer absent from its previously colored neighborhood. We prove that this rule can require $n^{Ω(1)}$ many colors on certain bipartite graph of $n$ vertices even when the vertices arrive in a uniformly random order. This refutes the conjecture of Bosek, Gutowski, Lasoń, and Przybyło [BGLP24] that FirstFit uses a polylogarithmic expected number of colors on every bipartite graph in the random order model. We obtain the lower bound by analyzing the FirstFit coloring process on random bipartite graphs, where vertices arrive to either side with equal probability, and with an independent edge density $n^{-x}$ for some $0 0$, there are graphs with chromatic number at most $n^ε$ on which FirstFit uses $Ω(n/\log n)$ colors in expectation under a uniformly random order. Our result shows that such poor behavior of FirstFit already occurs for $χ(G) = 2$.