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

TL;DR

This paper determines the smallest size of odd-girth critical graphs with respect to homomorphisms to odd cycles, generalizing classical critical graph concepts and providing exact values and computational classifications.

Contribution

It generalizes extremal questions for critical graphs to $H$-critical graphs with odd cycles, establishing exact minimal orders and characterizing specific critical graphs.

Findings

$ ext{eta}(k,C_{2 ext{l}+1})=4k$ for certain parameters

The smallest odd-girth 7-critical graph has 15 vertices

Exactly two of the 15-vertex graphs are $C_5$-critical

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.