Bounds on Odd and Odd-Even Induced Subgraphs

2026-08-03

AI Summary

This study addresses the problem of establishing lower bounds on the size of maximum induced subgraphs in graphs without isolated vertices, under vertex-degree parity constraints—either all odd or mixed parity. By generalizing Zeng’s odd-cut method, introducing a one-sided completion lemma, and combining an analysis of the $\mathbb{F}_2$-rank of bipartite adjacency matrices with fourth-moment inequalities, the authors derive improved bounds: $h_\ell(G) \geq n/6$ for general graphs and $f_o(G) \geq 65n/256$ for bipartite graphs, substantially surpassing the factor-of-two limitation inherent in Scott’s classical bound. Additionally, refined bounds depending on the independence number $\alpha$ are provided, and constructions demonstrate that the logarithmic improvements achieved are asymptotically optimal.

Abstract

Let $G$ be an $n$-vertex graph and let $\ell:V(G)\to\mathbb{F}_2$ prescribe degree parities. A set $S\subseteq V(G)$ is $\ell$-admissible if every $v\in S$ has degree congruent to $\ell(v)$ modulo $2$ in $G[S]$. Let $h_\ell(G)$ be the maximum order of an $\ell$-admissible set, set $f_{\mathrm{oe}}(G):=\min_\ell h_\ell(G)$, and write $f_o(G):=h_{\mathbf{1}}(G)$, where $\mathbf{1}(v)=1$ for every $v\in V(G).$ We prove three main results for graphs without isolated vertices. First, by extending Zeng's odd-cut method to arbitrary parity prescriptions an introducing a one-sided completion lemma, we show that $h_\ell(G)\ge n/6$ for every $\ell$. Consequently, $f_{\mathrm{oe}}(G)\ge n/6$, improving the previous bound $2n/21$. Second, for bipartite graphs we derive lower bounds on $f_o(G)$ in terms of the $\mathbb{F}_2$-rank of the bipartite adjacency matrix and combine them to obtain \[ f_o(G)\ge \left(\frac14+\frac1{256}\right)n=\frac{65}{256}n. \] Thus, in the bipartite case, the factor $2$ in Scott's bound $f_o(G)\ge n/(2χ(G))$ can be replaced by $128/65<2$. Finally, writing $α=α(G)$, a fourth-moment argument gives, for $α\ge2$, \[ f_o(G)\ge \fracα{2}+\frac{\log_3α}{8} -\frac14\log_3\log_3\sqrtα. \] We also construct bipartite graphs satisfying \[ f_o(G)\le \frac{α(G)}2+\log_2\!\bigl(α(G)+1\bigr)+\frac12, \] showing that the logarithmic additive improvement over Scott's bound $f_o(G)\geα(G)/2$ has the optimal order of magnitude.

Problem

• induced subgraph
• degree parity
• odd subgraph
• bipartite graph
• lower bound

Innovation

• odd-induced subgraphs
• parity constraints
• bipartite graphs
• F2-rank
• fourth-moment method