Graph Partitions Under Average Degree Constraint

TL;DR

This paper proves a conjecture that graphs with sufficiently high average degree can be partitioned into two parts with specified average degree constraints, advancing understanding of graph partitioning properties.

Contribution

It establishes a new theorem confirming that graphs with average degree at least s+t+2 can be partitioned to meet given average degree conditions, solving a longstanding conjecture.

Findings

Graphs with average degree ≥ s+t+2 can be partitioned into two parts with specified average degrees.

The conjecture by Csóka et al. is proven true.

The result extends the theory of graph partitions under degree constraints.

Abstract

In this paper, we prove that every graph with average degree at least $s+t+2$ has a vertex partition into two parts, such that one part has average degree at least $s$, and the other part has average degree at least $t$. This solves a conjecture of Cs\'{o}ka, Lo, Norin, Wu and Yepremyan.