Some results on the k-strong parity property in a graph

A graph $G$ has the $k$-strong parity property if for any $X\subseteq V(G)$ with $\mathrm{\mid }X\mathrm{\mid }$ even, $G$ contains a spanning subgraph $F$ with $d_F(u)\equiv 1$ (mod 2) for each $u\in X$ and $d_F(v)\in \{k,k+2,k+4,\dots \}$ for each $v\in V(G)\setminus X$, where $k\ge 2$ is an even integer. Kano and Matsumura proposed a characterization for a graph with the $k$-strong parity property (M. Kano, H. Matsumura, Odd-even factors of graphs, Graphs Combin. 41 (2025) 55). In this paper, we first give a size condition for a graph to have the $k$-strong parity property. Then we establish a signless Laplacian spectral radius condition to guarantee that a graph has the $k$-strong parity property.