Generalizations of Ferber-Krivelevich and Gallai Theorems on parity of degrees in induced subgraphs

2025-09-01

AI Summary

This paper investigates the induced subgraph parity problem for labeled vertices in graphs: (1) For undirected graphs without isolated vertices, it seeks a linear-size induced subgraph where vertices labeled 0 (resp. 1) have even (resp. odd) degree; (2) For directed graphs, it characterizes the necessary and sufficient conditions under which the vertex set admits a Gallai-type bipartition—i.e., a partition into two parts such that all vertices in the respective induced subgraphs have even and odd outdegrees. Innovatively introducing vertex labels to parity subgraph problems, the work provides the first nontrivial generalization of the Ferber–Krivelevich theorem. Using mod-2 linear algebra and vector space analysis, it fully characterizes the existence of such partitions via a rank condition on an associated incidence matrix over GF(2). It proves that every isolate-free undirected graph contains an induced subgraph satisfying the label-parity constraints with size Ω(n), and derives structural and matrix-rank characterizations for the class of directed graphs admitting such partitions.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Generalizing induced subgraph theorems with labeled vertex parity constraints
• Extending Gallai's Theorem to directed graphs with out-degree conditions
• Characterizing digraphs admitting bipartitions by parity of out-degrees

Innovation

• Generalized Ferber-Krivelevich theorem with vertex labeling
• Extended Gallai's Theorem to labeled parity conditions
• Characterized digraphs using linear algebraic methods