Bounds on Odd and Odd-Even Induced Subgraphs
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 $f_o(G)\ge n/(2\chi(G))$ can be replaced by $128/65<2$. Finally, writing $\alpha=\alpha(G)$, a fourth-moment argument gives, for $\alpha\ge2$, \[ f_o(G)\ge \frac{\alpha}{2}+\frac{\log_3\alpha}{8} -\frac14\log_3\log_3\sqrt{\alpha}. \] We also construct bipartite graphs satisfying \[ f_o(G)\le \frac{\alpha(G)}2+\log_2\!\bigl(\alpha(G)+1\bigr)+\frac12, \] showing that the logarithmic additive improvement over Scott's bound $f_o(G)\ge\alpha(G)/2$ has the optimal order of magnitude.