Some conditions for hamiltonian cycles in 1-tough $(K_2 \cup kK_1)$-free graphs

TL;DR

This paper proves a conjecture that certain 1-tough, k-connected, and $(K_2 ormation of the paper's core contribution in plain language.

Contribution

It establishes new conditions under which $(K_2 ormation of the paper's core contribution in plain language.

Findings

Proves that such graphs are Hamiltonian or the Petersen graph.

Identifies minimum degree conditions for Hamiltonicity.

Confirms the Shi and Shan conjecture for these graph classes.

Abstract

Let $k \geq 2$ be an integer. We say that a graph $G$ is $(K_2 \cup kK_1)$-free if it does not contain $K_2 \cup kK_1$ as an induced subgraph. Recently, Shi and Shan conjectured that every $1$-tough and $2k$-connected $(K_2 \cup kK_1)$-free graph is hamiltonian. In this paper, we solve this conjecture by proving the statement; every $1$-tough and $k$-connected $(K_2 \cup kK_1)$-free graph with minimum degree at least $\frac{3(k-1)}{2}$ is hamiltonian or the Petersen graph.