Smallest C-Critical Graphs of Odd-Girth 2k+1

Laurent Beaudou, Florent Foucaud, Reza Naserasr

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).