Eigenvalues and parity factors in graphs

TL;DR

This paper establishes optimal eigenvalue bounds in highly connected graphs to ensure the existence of specific parity-constrained spanning subgraphs, extending previous theoretical results in graph theory.

Contribution

It provides sharp eigenvalue bounds guaranteeing $(g,f)$-parity factors in graphs, extending earlier work and including optimality demonstrations.

Findings

Derived sharp eigenvalue bounds for parity factors

Extended previous theorems to broader graph classes

Provided graphs demonstrating bound optimality

Abstract

Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $g(v) \le d_H(v) \le f(v)$ and $f(v)\equiv d_H(v) \pmod{2}$. We prove sharp upper bounds for certain eigenvalues in an $h$-edge-connected graph $G$ with given minimum degree to guarantee the existence of a $(g,f)$-parity factor; we provide graphs showing that the bounds are optimal. This result extends the recent one of the second author (2022), extending the one of Gu (2014), Lu (2010), Bollb{\'a}s, Saito, and Wormald (1985), and Gallai (1950).