On the Density of $C_7$-Critical Graphs

Abstract

In 1959, Gr\"{o}tzsch 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 $4t$ admits a homomorphism to $C_{2t+1}$. This is in fact the planar dual of a well-known conjecture of Jaeger which states that every $4t$-edge-connected graph admits a modulo $(2t+1)$-orientation. Though Jaeger's original conjecture was disproved in 2018 by Han et al., Lovasz et al. showed that every $6t$-edge connected graph admits a modulo $(2t+1)$-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 $C_7$. 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 $G \not \in \{C_3, C_5\}$, then $e(G) \geq \tfrac{17v(G)-2}{15}$.