The Generic Circular Triangle-Free Graph

Abstract

In this paper, we introduce the generic circular triangle-free graph $\mathbb C_3$ and propose a finite axiomatization of its first order theory. In particular, our main results show that a countable graph $G$ embeds into $\mathbb C_3$ if and only if it is a $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free graph. As a byproduct of this result, we obtain a geometric characterization of finite $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free graphs, and the (finite) list of minimal obstructions of unit Helly circular-arc graphs with independence number strictly less than three. The circular chromatic number $\chi_c(G)$ is a refinement of the classical chromatic number $\chi(G)$. We construct $\mathbb C_3$ so that a graph $G$ has circular chromatic number strictly less than three if and only if $G$ maps homomorphically to $\mathbb C_3$. We build on our main results to show that $\chi_c(G) < 3$ if and only if $G$ can be extended to a $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free graph, and in turn, we use this result to reprove an old characterization of $\chi_c(G) < 3$ due to Brandt (1999). Finally, we answer a question recently asked by Guzm\'an-Pro, Hell, and Hern\'andez-Cruz by showing that the problem of deciding for a given finite graph $G$ whether $\chi_c(G) < 3$ is NP-complete.