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Abstract

A long-standing and well-known conjecture (see e.g. Caro, Discrete Math, 1994) states that every $n$-vertex graph $G$ without isolated vertices contains an induced subgraph where all vertices have an odd degree and whose order is linear in $n$. Ferber and Krivelevich (Adv. Math., 2022) confirmed the conjecture. In this short paper, we generalize this result by considering $G$ with vertices labeled 0 or 1 and requiring that in an induced subgraph of $G$, the 0-labeled vertices are of even degree and the 1-labeled vertices are of odd degree. We prove that if $G$ has no isolated vertices, it contains such a subgraph of order linear in $n$. The well-known Gallai's Theorem states that the vertices of each graph can be partitioned into two parts such that all vertices in the subgraphs induced by the two parts have even degrees. The result also holds if we require that the degrees of all vertices in one of the induced subgraphs are even, and the degrees of all vertices in the other induced subgraph are odd. A natural generalization of Gallai's Theorem to out-degrees in digraphs does not hold and we characterize all digraphs for which it does hold. Our characterization is linear algebraic.