Sufficient spectral conditions for graphs being k-edge-Hamiltonian or k-Hamiltonian

Abstract

A graph G is k-edge-Hamiltonian if any collection of vertex-disjoint paths with at most k edges altogether belong to a Hamiltonian cycle in G. A graph G is k-Hamiltonian if for all $S\subseteq V\left(G\right)$ with $|S|\le k$, the subgraph induced by $V\left(G\right)\setminus S$ has a Hamiltonian cycle. These two concepts are classical extensions of the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be k-edge-Hamiltonian and k-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results could be viewed as slight extensions of the recent theorems proved by Li and Ning [Linear Multilinear Algebra. 2016;64:2252–2269], Nikiforov [Czechoslovak Math J. 2016;66:925–940] and Li et al. [Linear Multilinear Algebra. 2018;66:2011–2023]. Moreover, we shall prove a stability result for graphs being k-Hamiltonian, which could be regarded as a complement of two recent results of Füredi et al. [Discrete Math. 2017;340:2688–2690] and [Discrete Math. 2019;342:1919–1923].

Keywords:

• Spectral radius
• Hamiltonian cycle
• extremal graph theory
• stability

2010 Mathematics Subject Classifications:

• 05C50
• 15A18
• 05C38

Acknowledgments

We would like to thank the anonymous referees for their careful reviews.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (NSFC) [grant numbers 11931002 11671124].