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

TL;DR

This paper establishes a sharp spectral radius threshold for connected graphs to contain a parity [a,b]-factor, extending eigenvalue conditions for such factors and characterizing extremal graphs.

Contribution

It provides a new spectral radius bound ensuring the existence of parity [a,b]-factors in graphs, generalizing previous eigenvalue conditions and identifying extremal structures.

Findings

Spectral radius threshold guarantees parity [a,b]-factor existence

Characterization of extremal graphs not having such factors

Extension of eigenvalue conditions for regular graphs

Abstract

Let $a$, $b$, and $n$ be three integers such that $1\leq a \leq b < n$, $a \equiv b$ (mod $2$), and $na$ is even. A parity $[a,b]$-factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $a \leq d_H(v) \leq b$ 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 $\delta(G)\geq a$ and $\rho(G)\geq\rho(G_{n}^{a})$, then $G$ contains a parity $[a,b]$-factor unless $G \cong G_{n}^{a}$, where $2\leq a<b$ 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 $(a+1)K_1$.