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

TL;DR

This paper investigates the $k$-strong parity property in graphs, a generalization of the strong parity property, by Kano and Matsumura. It establishes two practical sufficient conditions: (i) a size-based condition (Theorem $1.2$) for connected graphs with minimum degree at least $k+1$, and (ii) a signless Laplacian spectral radius condition (Theorem $1.3$) guaranteeing the property. The proofs leverage extremal-edge-count arguments and a pivotal edge-comparison lemma, along with a standard spectral bound on the signless Laplacian. Together, these results advance understanding of when graphs admit spanning subgraphs with prescribed parity degrees, offering tools for both theory and potential applications in graph factor problems.

Abstract

A graph $G$ has the $k$-strong parity property if for any $X\subseteq V(G)$ with $|X|$ even, $G$ contains a spanning subgraph $F$ with $d_F(u)\equiv1$ (mod 2) for each $u\in X$ and $d_F(v)\in\{k,k+2,k+4,\ldots\}$ for each $v\in V(G)\setminus X$, where $k\geq2$ 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.