A long-standing conjecture of Caro (Discrete Math, 1994), confirmed by Ferber and Krivelevich (Adv Math, 2022), states that every $n$-vertex graph $G$ without isolated vertices contains an induced subgraph of order linear in $n$ in which every vertex has odd degree. We generalize this result to graphs $G$ whose vertices are labeled by $\ell: V(G)\to \{0,1\}$. We require, in an induced subgraph, all $0$-labeled vertices to have even degree and all $1$-labeled vertices to have odd degree. Let $h_{\ell}(G)$ denote the maximum order of such a subgraph. Let $f_{oe}(G)=\min_{\ell} h_{\ell}(G)$ be the worst-labeling parameter. We establish a pointwise lower bound for $h_{\ell}(G)$ that immediately yields a linear lower bound in $|V(G)|$ for $f_{oe}(G)$, where $G$ has no isolated vertices. For an $n$-vertex connected graph, we obtain a sharp lower bound for $f_{oe}(G)$: $f_{oe}(G)\ge \lceil (n-1)/χ_{mm}{(G)} \rceil ,$ where $χ_{mm}{(G)}$ is the maximum chromatic number of a minor of $G.$ Using proved cases of Hadwiger's Conjecture, we show that for $t\in \{3,4,5,6\}$, if an $n$-vertex connected graph $G$ is $K_t$-minor-free, then $f_{oe}(G)\ge \lceil (n-1)/(t-1)\rceil$ and this bound is sharp for each $t\in \{3,4,5,6\}$. Finally, we conjecture that $f_{oe}(G)\ge f_o(G)/2$ for all graphs $G$ and confirm the conjecture for all trees and complete multipartite graphs.

翻译：Caro（Discrete Math, 1994）提出的一个长期未解决的猜想（由Ferber和Krivelevich（Adv Math, 2022）证实）指出：每个无孤立顶点的$n$阶图$G$包含一个顶点数为$n$的线性阶诱导子图，其中每个顶点均具有奇度数。我们将此结果推广到顶点被标记为$\ell: V(G)\to \{0,1\}$的图$G$。我们要求诱导子图中所有标记为$0$的顶点具有偶度数，所有标记为$1$的顶点具有奇度数。设$h_{\ell}(G)$表示此类子图的最大阶数，并定义最坏标记参数$f_{oe}(G)=\min_{\ell} h_{\ell}(G)$。我们建立了$h_{\ell}(G)$的点态下界，该下界直接给出了$f_{oe}(G)$在$|V(G)|$上的线性下界（其中$G$无孤立顶点）。对于$n$阶连通图，我们得到了$f_{oe}(G)$的锐利下界：$f_{oe}(G)\ge \lceil (n-1)/χ_{mm}{(G)} \rceil$，其中$χ_{mm}{(G)}$是$G$的某个子式的最大色数。利用Hadwiger猜想已证明的情形，我们证明：对于$t\in \{3,4,5,6\}$，若$n$阶连通图$G$不含$K_t$子式，则$f_{oe}(G)\ge \lceil (n-1)/(t-1) \rceil$，且此界对每个$t\in \{3,4,5,6\}$都是锐利的。最后，我们猜想对所有图$G$有$f_{oe}(G)\ge f_o(G)/2$，并证实该猜想对所有树和完全多部图成立。