On the Density of C₇-Critical Graphs

Abstract

In 1959, Grötzsch [5] famously proved that every planar graph of girth at least 4 is 3-colourable (or equivalently, admits a homomorphism to C₃). A natural generalization of this is the following conjecture: for every positive integer t, every planar graph of girth at least 4t admits a homomorphism to C_2t+1. This is in fact the planar dual of a well-known conjecture of Jaeger [7] which states that every 4t-edge-connected graph admits a modulo (2t + 1)-orientation. Though Jaeger’s original conjecture was disproved in [6], Lovász et al. [10] showed that every 6t-edge connected graph admits a modolo (2t + 1)-flow. The latter result implies that every planar graph of girth at least 6t admits a homomorphism to C_2t+1. We improve upon this in the t = 3 case, by showing that every planar graph of girth at least 16 admits a homomorphism to C₇. We obtain this through a more general result regarding the density of C₇-critical graphs: if G is a C₇-critical graph with G ∉ {C₃, C₅}, then $e(G) \ge {{17v(G) - 2} \over {15}}$.