A Ramsey-type theorem on deficiency

TL;DR

This paper establishes a Ramsey-type theorem characterizing forbidden induced subgraphs for graphs with bounded deficiency, advancing understanding of graph structure related to matchings and answering a question from prior research.

Contribution

The paper proves a new Ramsey-type theorem for deficiency, identifying all forbidden induced subgraphs for graphs with bounded deficiency.

Findings

Characterization of forbidden induced subgraphs for bounded deficiency

Answer to a question posed by Fujita et al. (2006)

Extension of Ramsey's theorem to deficiency in graphs

Abstract

Ramsey's Theorem states that a graph $G$ has bounded order if and only if $G$ contains no complete graph $K_n$ or empty graph $E_n$ as its induced subgraph. The Gy\'arf\'as-Sumner conjecture says that a graph $G$ has bounded chromatic number if and only if it contains no induced subgraph isomorphic to $K_n$ or a tree $T$. The deficiency of a graph is the number of vertices that cannot be covered by a maximum matching. In this paper, we prove a Ramsey type theorem for deficiency, i.e., we characterize all the forbidden induced subgraphs for graphs $G$ with bounded deficiency. As an application, we answer a question proposed by Fujita, Kawarabayashi, Lucchesi, Ota, Plummer and Saito (JCTB, 2006).