Spectral radius and parity $[a,b]$-factors in graphs

Abstract

Let $a$, $b$, and $n$ be three integers such that $1\le a\le b<n$, $a\equiv b$ (mod $2$), and $na$ is even. A parity $[a,b]$-factor of $G$ is a spanning subgraph $H$ such that for each vertex $v\in V(G)$, $a\le d_H(v)\le b$ and $d_H(v)\equiv a\equiv b$ (mod $2$). Recently, O [J. Graph Theory 100 (2022) 458-469] proved eigenvalue conditions for a regular graph to have a parity $[a,b]$-factor. In this paper, we prove a sharp lower bound on the spectral radius for an $n$-vertex graph $G$ to have a parity $[a,b]$-factor as follows: If $G$ is an $n$-vertex connected graph with $\hat{I}´(G)\ge a$ and $\ddot{I}(G)\ge \ddot{I}(G_n^a)$, then $G$ contains a parity $[a,b]$-factor unless $G\cong G_n^a$, where $2\le a<b$ and $G_n^a$ is the graph obtained from $K_{a-1}\vee (K_{n-2a-1}\cup (a+1)K_1)$ by adding a new vertex and adding all possible edges between the added vertex and each vertex in $(a+1)K_1$.

18 pages, 1 figure