On Vertex Bisection Width of Random $d$-Regular Graphs

TL;DR

This paper establishes the first known upper bounds on the vertex bisection width of random $d$-regular graphs for constant $d$, using a differential equations approach to analyze a greedy algorithm.

Contribution

It introduces a novel application of the Differential Equations Method to derive upper bounds for vertex bisection width in random regular graphs.

Findings

First known upper bounds for vertex bisection width in random regular graphs.

Comparison with experimental results and existing lower bounds.

Insights into the effectiveness of the greedy algorithm for graph partitioning.

Abstract

Vertex bisection is a graph partitioning problem in which the aim is to find a partition into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. We are interested in giving upper bounds on the vertex bisection width of random $d$-regular graphs for constant values of $d$. Our approach is based on analyzing a greedy algorithm by using the Differential Equations Method. In this way, we obtain the first known upper bounds for the vertex bisection width in random regular graphs. The results are compared with experimental ones and with lower bounds obtained by Kolesnik and Wormald, (Lower Bounds for the Isoperimetric Numbers of Random Regular Graphs, SIAM J. on Disc. Math. 28(1), 553-575, 2014).