### Book chapter

## Smallest πΆ_{2π+1}-Critical Graphs of Odd-Girth 2π+1

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 *πΎ**π*−1-critical graphs are the classic *k*-(colour)-critical graphs. This work is a first step towards extending questions of extremal nature from *k*-critical graphs to *H*-critical graphs. Besides complete graphs, the next classic case is odd cycles. Thus, given integers *π*≥*π* we ask: what is the smallest order *π*(*π*,*π*) of a *πΆ*2*π*+1-critical graph of odd-girth at least 2*π*+1? Denoting this value by *π*(*π*,*π*), we show that *π*(*π*,*π*)=4*π* for *π*≤*π*≤3*π*+*π*−32 (2*π*=*π*mod3) and that *π*(3,2)=15. The latter is to say that a smallest graph of odd-girth 7 not admitting a homomorphism to the 5-cycle is of order 15 (there are at least 10 such graphs on 15 vertices).