Degree-equipartite graphs

TL;DR

This paper characterizes all degree-equipartite graphs, which are graphs where any partition of vertices into two equal parts results in subgraphs with identical degree sequences, addressing a previously open problem.

Contribution

It provides a complete characterization of degree-equipartite graphs, solving an open problem posed by Grünbaum et al.

Findings

All degree-equipartite graphs are characterized explicitly.

The characterization confirms the structure of such graphs.

The result resolves an open problem in graph theory.

Abstract

A graph $G$ of order $2n$ is called degree-equipartite if for every $n$-element set $A\subseteq V(G)$, the degree sequences of the induced subgraphs $G[A]$ and $G[V(G)\setminus A]$ are the same. In this paper, we characterize all degree-equipartite graphs. This answers Problem 1 in the paper by Gr\"{u}nbaum et al [B. Gr\"{u}nbaum, T. Kaiser, D. Kr\'{a}l, and M. Rosenfeld, Equipartite graphs, {\it Israel J. Math.} {\bf 168} (2008), 431-444].