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

TL;DR

This paper establishes spectral conditions based on adjacency and signless Laplacian spectra that guarantee a graph's $k$-edge-Hamiltonian or $k$-Hamiltonian property, extending previous theorems and including stability results.

Contribution

It provides new spectral criteria for $k$-edge-Hamiltonian and $k$-Hamiltonian graphs, extending recent theoretical results and offering stability analysis.

Findings

Spectral conditions for $k$-edge-Hamiltonian graphs

Spectral conditions for $k$-Hamiltonian graphs

Stability results for $k$-Hamiltonian graphs

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 $S\subseteq V(G)$ with $|S|\le k$, the subgraph induced by $V(G)\setminus S$ has a Hamiltonian cycle. These two concepts are classical extensions for 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 64 (2016)], Nikiforov [Czechoslovak Math. J. 66 (2016)] and Li, Liu and Peng [Linear Multilinear Algebra 66 (2018)]. Moreover, we shall prove a stability result for graphs being $k$-Hamiltonian, which could be regarded as a complement of two recent results of F\"{u}redi, Kostochka and Luo [Discrete Math. 340 (2017)] and [Discrete Math. 342 (2019)].