Large homeomorphically irreducible trees in path‐free graphs

Abstract

AbstractA subgraph of a graph is a homeomorphically irreducible tree (HIT) if it is a tree with no vertices of degree two. Furuya and Tsuchiya [Discrete Math. 313 (2013), pp. 2206–2212] proved that every connected ‐free graph of order has a HIT of order at least , and Diemunsch et al [Discrete Appl. Math. 185 (2015), pp. 71–78] proved that every connected ‐free graph of order has a HIT of order at least . In this paper, we continue the study of HITs in path‐free graphs and bring to a conclusion proving the following three results: (a) If a connected graph satisfies the condition that every connected ‐free graph has a HIT of order , then is a path of order at most . (b) Every connected ‐free graph of order has a HIT of order at least . (c) Every connected ‐free graph of order has a HIT of order at least . Furthermore, we focus on a ‐free graph having large minimum degree, and prove that for a connected ‐free graph of order , if , then has a HIT of order greater than .