Mycielski Graphs and [Formula: see text] Proofs

Yolcu, Emre; Wu, Xinyu; Heule, Marijn J. H.

Mycielski graphs are a family of triangle-free graphs [Formula: see text] with arbitrarily high chromatic number. [Formula: see text] has chromatic number k and there is a short informal proof of this fact, yet finding proofs of it via automated reasoning techniques has proved to be a challenging task. In this paper, we study the complexity of clausal proofs of the uncolorability of [Formula: see text] with [Formula: see text] colors. In particular, we consider variants of the [Formula: see text] (propagation redundancy) proof system that are without new variables, and with or without deletion. These proof systems are of interest due to their potential uses for proof search. As our main result, we present a sublinear-length and constant-width [Formula: see text] proof without new variables or deletion. We also implement a proof generator and verify the correctness of our proof. Furthermore, we consider formulas extended with clauses from the proof until a short resolution proof exists, and investigate the performance of CDCL in finding the short proof. This turns out to be difficult for CDCL with the standard heuristics. Finally, we describe an approach inspired by SAT sweeping to find proofs of these extended formulas.