The existence of odd-even factors in 1-binding graphs

Abstract

Let $G$ be a graph. The binding number of $G$, denoted by $\mbox{bind}(G)$, is defined as $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(S)|}{|S|}:\emptyset\neq S\subseteq V(G) \ \mbox{and} \ N_G(S)\neq V(G)\right\}. $$ If $\mbox{bind}(G)\geq r$, then $G$ is called $r$-binding, where $r$ is a positive real number. The adjacency matrix of $G$ is denoted by $A(G)$. The largest eigenvalue of $A(G)$, denoted by $\rho(G)$, is said to be the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an odd-even factor $F=F_W$ if $d_F(u)\in\{1,3,\ldots,k\}$ for every $u\in W$ and $d_F(v)\in\{0,2,\ldots,k+1\}$ for every $v\in V(G)-W$, where $k$ is a positive odd integer and $W$ is any set of even number of vertices of $G$. In this paper, we propose a tight sufficient condition based on the spectral radius to guarantee that a connected 1-binding graph $G$ contains an odd-even factor $F=F_W$ such that $d_F(u)\in\{1,3,\ldots,k\} \ \mbox{for all} \ u\in W$ and $d_F(v)\in\{0,2,\ldots,k+1\} \ \mbox{for all} \ v\in V(G)-W$.