Prime graphs, matchings and the Castelnuovo-Mumford regularity

Abstract

We demonstrate the effectiveness of prime graphs for the calculation of the (Castelnuovo-Mumford) regularity of graphs. Such a notion allows us to reformulate the regularity as a generalized induced matching problem and perform regularity calculations in specific graph classes, including $(C_3,P_5)$-free graphs, $P_6$-free bipartite graphs and all Cohen-Macaulay graphs of girth at least five. In particular, we verify that the five cycle graph $C_5$ is the unique connected graph satisfying the inequality $im (G)\lt \mbox {reg}(G)=m (G)$. In addition, we prove that, for each integer $n\geq 1$, there exists a vertex decomposable perfect prime graph $G_n$ with $\mbox {reg}(G_n)=n$.