On the Density of C7-Critical Graphs Luke Postle, Evelyne Smith-Roberge

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