Density of 5/2-critical graphs

Abstract

A graph G is 5/2-critical if G has no circular 5/2-coloring (or equiva-lently, homomorphism to C5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n ≥ 4 vertices has at least 5n−24 edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.