Density and Structure of Homomorphism-Critical Graphs

Abstract

Let $H$ be a graph. A graph $G$ is $H$-critical if every proper subgraph of $G$ admits a homomorphism to $H$, but $G$ itself does not. In 1981, Jaeger made the following conjecture concerning odd-cycle critical graphs: every planar graph of girth at least $4t$ admits a homomorphism to $C_{2t+1}$ (or equivalently, has a $\tfrac{2t+1}{t}$-circular colouring). The best known result for the $t=3$ case states that every planar graph of girth at least 18 has a homomorphism to $C_7$. We improve upon this result, showing that every planar graph of girth at least 16 admits a homomorphism to $C_7$. This is obtained from a more general result regarding the density of $C_7$-critical graphs. Our main result is that if $G$ is a $C_7$-critical graph with $G \not \in \{C_3, C_5\}$, then $e(G) \geq \tfrac{17v(G)-2}{15}$. Additionally, we prove several structural lemmas concerning graphs that are $H$-critical, when $H$ is a vertex-transitive non-bipartite graph.