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

Let $G$ be a graph. The binding number of $G$, denoted by $bind(G)$, is defined as $$bind(G)=\min \{\frac{\mathrm{\mid }{N}_G(S)\mathrm{\mid }}{\mathrm{\mid }S\mathrm{\mid }}:\mathrm{\varnothing }\mathrm{\ne }S\subseteq V(G)and{N}_G(S)\mathrm{\ne }V(G)\}\mathrm{.}$$ If $bind(G)\ge r$, 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 $d_F(u)\in \{1,3,\dots ,k\}$ for every $u\in W$ and $d_F(v)\in \{0,2,\dots ,k+1\}$ 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 $d_F(u)\in \{1,3,\dots ,k\}forallu\in W$ and $d_F(v)\in \{0,2,\dots ,k+1\}forallv\in V(G)-W$.