Spectral radius and parity $[a,b]$-factors in graphs
Ruifang Liu, Ting Xu, Suil O
TL;DR
We address sharp spectral conditions guaranteeing parity $[a,b]$-factors in connected graphs with minimum degree at least $a$. The authors derive a tight threshold $\rho(G)\ge \rho(G_n^{a})$ ensuring the existence of a parity $[a,b]$-factor, except for the explicitly constructed extremal graph $G_n^{a}$, defined via a join and a pendant-structure around a new vertex. The proof combines Lovasz parity-factor criterion with spectral-radius bounds and equitable-partition techniques to characterize the extremal graphs and establish the bound as tight. This work extends prior results for regular graphs and provides a precise extremal construction for parity $[a,b]$-factors in terms of a concrete extremal graph $G_n^{a}$.
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 $δ(G)\geq a$ and $ρ(G)\geqρ(G_{n}^{a})$, 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 $(a+1)K_1$.