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 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 $l\ge k$ we ask: what is the smallest order $\eta (k,l)$ of a $C_{2l+1}$-critical graph of odd-girth at least $2k+1$? Denoting this value by $\eta (k,l)$, we show that $\eta (k,l)=4k$ for $l\le k\le \frac{3l+i-3}{2}$ ($2k=i\bmod 3$) and that $\eta (3,2)=15$. 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).

This work is supported by the IFCAM project Applications of graph homomorphisms (MA/IFCAM/18/39) and by the ANR project HOSIGRA (ANR-17-CE40-0022).