Forbidden subgraphs and 2‐factors in 3/2‐tough graphs

Abstract

A graph G is H -free if it has no induced subgraph isomorphic to H , where H is a graph. In this paper, we show that every 3 2 -tough ∪ P P ( ) 4 1 0 -free graph has a 2-factor. The toughness condition of this result is sharp. Moreover, for any ε > 0 there exists a ε (2 − )tough P 2 5 -free graph without a 2-factor. This implies that the graph ∪ P P 4 1 0 is best possible for a forbidden subgraph in a sense. K E Y W O R D S 2-factor, Hamiltonian cycle, (P 4 ∪ P 10 )-free graph, toughness Y to denote ∩ Y N X ( ) G . If ≥ e x Y ( , ) 1 G ≥ e X Y ( ( , ) 1) G , we say that x and Y (X and Y ) are adjacent.

Comparison of Contents Design in Mathematical Inquiry in High School Textbooks

“…This implies that every 3 2 -tough 2P 2free graph on at least three vertices has a 2-factor, see [10]. Sanka [12] proved that every 3 2 -tough (P 10 ∪ P 4 )-free graph has a 2-factor. Grimm, Johnsen and Shan [7] found sharp bound less than 2 on t such that every t-tough R-free graphs with at least three vertices has a 2-factor for any forest R on 5, 6, 7 vertices.…”