Adjacency spectral radius and H-factors in 1-binding graphs

TL;DR

This paper establishes a spectral radius condition that guarantees the existence of certain $H$-factors in 1-binding graphs, extending previous characterizations and providing new sufficient conditions based on adjacency spectral properties.

Contribution

It introduces a spectral radius criterion for the existence of $H$-factors in 1-binding graphs, generalizing prior results and characterizing extremal cases.

Findings

If $ ho(G) extgreater{} ho(K_1igvee(K_{n-4}igcup K_2igcup K_1))$, then $G$ has an $H$-factor for specified $H$.

The result applies to connected 1-binding graphs of order at least 11.

The paper characterizes the extremal graph where the spectral radius condition is tight.

Abstract

Let $G$ be a graph, and let $H:V(G)\longrightarrow\{\{1\},\{0,2\}\}$ be a set-valued function. Hence, $H(v)$ equals $\{1\}$ or $\{0,2\}$ for any $v\in V(G)$. We let $$ H^{-1}(1)=\{v: v\in V(G) \ \mbox{and} \ H(v)=1\}. $$ An $H$-factor of $G$ is a spanning subgraph $F$ of $G$ such that $d_F(v)\in H(v)$ for each $v\in V(G)$. Lu and Kano showed a characterization for the existence of an $H$-factor in a graph [Characterization of 1-tough graphs using factors, Discrete Math. 343 (2020) 111901]. Let $A(G)$ and $\rho(G)$ denote the adjacency matrix and the adjacency spectral radius of $G$, respectively. By using Lu and Kano's result, we pose a sufficient condition with respect to the adjacency spectral radius to guarantee the existence of an $H$-factor in a 1-binding graph. In this paper, we prove that if a connected 1-binding graph $G$ of order $n\geq11$ satisfies $\rho(G)\geq\rho(K_1\vee(K_{n-4}\cup K_2\cup K_1))$, then $G$ has an $H$-factor for each $H:V(G)\longrightarrow\{\{1\},\{0,2\}\}$ with $H^{-1}(1)$ even, unless $G=K_1\vee(K_{n-4}\cup K_2\cup K_1)$.