Equating three degrees of graphs

TL;DR

This paper proves that any graph with at least five vertices can be reduced by removing at most three vertices to ensure the remaining subgraph has at least three vertices sharing the same degree, solving an open problem.

Contribution

It establishes a new vertex deletion bound to achieve degree repetition in subgraphs, resolving an open problem in graph theory.

Findings

Any graph with ≥5 vertices can be reduced to have three vertices with the same degree by removing ≤3 vertices.

The result confirms a conjecture posed by Caro, Shapira, and Yuster.

The proof provides a new insight into degree distributions in subgraphs.

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].