Hamiltonian Cycles in Tough $(P_2\cup P_3)$-Free Graphs

Abstract

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chv\'atal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. A graph is called $(P_2\cup P_3)$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, respectively. In this paper, we show that every 15-tough $(P_2\cup P_3)$-free graph with at least three vertices is hamiltonian.

An Ore-Type Condition for Hamiltonicity in Tough Graphs

“…Since H has order λ and no vertex of H is adjacent in G to any vertex of v + i Cx − or v + j Cy − by the first part of the statement, it follows that each component of The following lemma provides a way of extending a cycle C provided that the vertices outside C have many neighbors on C. The proof follows from Lemma 8 and is very similar to the proof of Lemma 10 in [10].…”

An Ore-type condition for hamiltonicity in tough graphs

“…The fact below was used in many papers when finding a long cycle under a given toughness condition, and a proof can be found in [10].…”

“…The following lemma provides a way of extending a cycle C longer provided that the vertices outside C have many neighbors on C. In [10], the result was proved using the condition deg G (x, C) > n t+1 , but the proof follows exactly the same argument. Lemma 3.…”

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

“…It has been shown that Conjecture 1.1 is true for some superclass of split graphs, for example, spider graphs [11], chordal graphs [6,10], 2K 2 -free graphs [5,15,14], and (P 2 ∪ P 3 )-free graphs [16]. However, some of the above results are not known to be best about the toughness condition, which cannot be smaller than 3 2 by Theorem 1.2.…”