On vertex decomposable and Cohen-Macaulay regular graphs

Abstract

We characterize the Cohen-Macaulay property for generalized Petersen graphs and $3$-regular graphs. In particular, we prove that these graphs are vertex decomposable. Also, we characterize pure vertex decomposability for $4$-transitive graphs without $5$-holes. Finally, we study the small cycles of well-covered and Cohen-Macaulay regular graphs.