Smallest $C_{2l+1}$-critical graphs of odd-girth $2k+1$

Abstract

Given a graph $H$, a graph $G$ is called $H$-critical if $G$ does not admit a homomorphism to $H$, but any proper subgraph of $G$ does. Observe that $K_{k-1}$-critical graphs are the standard $k$-(colour)-critical graphs. We consider questions of extremal nature previously studied for $k$-critical graphs and generalize them to $H$-critical graphs. After complete graphs, the next natural case to consider for $H$ is that of the odd-cycles. Thus, given integers $\ell$ and $k$, $\ell\geq k$, we ask: what is the smallest order of a $C_{2\ell +1}$-critical graph of odd-girth at least $2k+1$? Denoting this value by $\eta(k,C_{2\ell+1})$, we show that $\eta(k,C_{2\ell+1})=4k$ for $1\leq\ell\leq k\leq\frac{3\ell+i-3}{2}$ ($2k=i\bmod 3$) and that $\eta(3,C_5)=15$. The latter means that a smallest graph of odd-girth~$7$ not admitting a homomorphism to the $5$-cycle is of order~$15$. Computational work shows that there are exactly eleven such graphs on $15$~vertices of which only two are $C_5$-critical.