On the chromatic number of random triangle-free graphs

Abstract

We study the chromatic number of typical triangle-free graphs with $\Theta \left( n^{3/2} (\log n)^{1/2} \right)$ edges and establish the width of the scaling window for the transitions from $\chi = 3$ to $\chi = 4$ and from $\chi = 4$ to $\chi = 5$. The transition from $3$- to $4$-colorability has scaling window of width $\Theta(n^{4/3} (\log n)^{-1/3})$. To prove this, we show a high probability equivalence of the $3$-colorability of a random triangle-free graph at this density and the satisfiability of an instance of bipartite random $2$-SAT, for which we establish the width of the scaling window following the techniques of Bollob{\'a}s, Borgs, Chayes, Kim, and Wilson. The transition from $4$- to $5$-colorability has scaling window of width $\Theta(n^{3/2} (\log n)^{-1/2})$. To prove this, we show a high probability equivalence of the $4$-colorability of a random triangle-free graph at this density and the simultaneous $2$-colorability of two independent Erd\H{o}s--R\'enyi random graphs. For this transition, we also establish the limiting probability of $4$-colorability inside the scaling window.