Fair splittings by independent sets in sparse graphs

TL;DR

This paper establishes topological conditions for partitioning sparse graphs into multiple disjoint independent sets that accurately reflect given vertex distributions, extending combinatorial graph theory with topological methods.

Contribution

It introduces a topological framework linking independence complexes to the existence of multiple disjoint independent sets with specified vertex proportions in sparse graphs.

Findings

Existence of multiple disjoint independent sets under certain topological conditions.

Application of equivariant topology to graph partition problems.

Results hold for graphs with sparse structures and prime power parameters.

Abstract

Given a partition $V_1 \sqcup V_2 \sqcup \dots \sqcup V_m$ of the vertex set of a graph, we are interested in finding multiple disjoint independent sets that contain the correct fraction of vertices of each $V_j$. We give conditions for the existence of $q$ such independent sets in terms of the topology of the independence complex. We relate this question to the existence of $q$-fold points of coincidence for any continuous map from the independence complex to Euclidean space of a certain dimension, and to the existence of equivariant maps from the $q$-fold deleted join of the independence complex to a certain representation sphere of the symmetric group. As a corollary we derive the existence of $q$ pairwise disjoint independent sets accurately representing the $V_j$ in certain sparse graphs for $q$ a power of a prime.