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Equating three degrees of graphs

Zhen Liu, Qinghou Zeng

TL;DR

The paper resolves the exact value of $C(3)$ for general graphs by proving that every $n$-vertex graph with $n\ge5$ contains an induced subgraph formed by deleting at most $3$ vertices in which at least three vertices share the same degree, i.e., $C(3)=3$. The approach relies on a framework of feasible sets, balanceable and accessible configurations, supported by the Sun–Hou–Zeng lemmas, and a key four-vertex degree-pattern lemma to control local structures. The argument proceeds via a contradiction on degree multisets using sets $S$ and $T$ and modular considerations, ultimately ruling out counterexamples. The paper also outlines open problems for larger targets $C(k)$ and connects to the related degree-equating problem for maximum degree via $f_k(G)$ and $f_k(n)$, pointing to future research directions.

Abstract

In this paper, we prove that, for every graph with at least 5 vertices, one can delete at most 3 vertices such that the subgraph obtained has at least three vertices with the same degree. This solves an open problem of Caro, Shapira and Yuster [Electron. J. Combin. 21 (2014) P1.24].

Equating three degrees of graphs

TL;DR

The paper resolves the exact value of for general graphs by proving that every -vertex graph with contains an induced subgraph formed by deleting at most vertices in which at least three vertices share the same degree, i.e., . The approach relies on a framework of feasible sets, balanceable and accessible configurations, supported by the Sun–Hou–Zeng lemmas, and a key four-vertex degree-pattern lemma to control local structures. The argument proceeds via a contradiction on degree multisets using sets and and modular considerations, ultimately ruling out counterexamples. The paper also outlines open problems for larger targets and connects to the related degree-equating problem for maximum degree via and , pointing to future research directions.

Abstract

In this paper, we prove that, for every graph with at least 5 vertices, one can delete at most 3 vertices such that the subgraph obtained has at least three vertices with the same degree. This solves an open problem of Caro, Shapira and Yuster [Electron. J. Combin. 21 (2014) P1.24].
Paper Structure (4 sections, 5 theorems, 13 equations, 1 figure)

This paper contains 4 sections, 5 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Feasible sets and related lemmas
  3. Proof of Theorem \ref{['Delete3']}
  4. Concluding remarks

Key Result

Theorem 1.2

For any graph $G$ with at least $5$ vertices, one can delete at most $3$ vertices such that the subgraph obtained has at least three vertices with the same degree. Consequently, $C(3)=3$.

Figures (1)

  • Figure 1: $C(3)\geq 3$

Theorems & Definitions (13)

  • Theorem 1.2
  • Lemma 2.1: Sun, Hou and Zeng SHZ
  • Lemma 2.2: Sun, Hou and Zeng SHZ
  • Lemma 2.3: Sun, Hou and Zeng SHZ
  • Lemma 2.4
  • proof : Proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['Delete3']}
  • proof
  • ...and 3 more