Relating the independence number and the dissociation number

Abstract

The independence number α(G) and the dissociation number diss(G) of a graph G are the largest orders of induced subgraphs of G of maximum degree at most 0 and at most 1, respectively. We consider possible improvements of the obvious inequality 2α(G) ≥ diss(G). For connected cubic graphs G distinct from K 4 , we show 5α(G) ≥ 3diss(G), and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle-free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.