On induced saturation for paths

Abstract

For a graph H, a graph G is H-induced-saturated if G does not contain an induced copy of H, but either removing an edge from G or adding a non-edge to G creates an induced copy of H. Depending on the graph H, an H-induced-saturated graph does not necessarily exist. In fact, Martin and Smith [11] showed that P 4 -induced-saturated graphs do not exist, where P k denotes a path on k vertices. Axenovich and Csikós [1] asked the existence of P k -induced-saturated graphs for k ≥ 5; it is easy to construct such graphs when k ∈ {2, 3}. Recently, Räty [13] constructed a graph that is P 6 -induced-saturated.In this paper, we show that there exists a P k -induced-saturated graph for infinitely many values of k. To be precise, we find a P 3n -induced-saturated graph for every positive integer n. As a consequence, for each positive integer n, we construct infinitely many P 3n -induced-saturated graphs. We also show that the Kneser graph K(n, 2) is P 6 -induced-saturated for every n ≥ 5.