A Generalization of Gale's lemma

TL;DR

This paper generalizes Gale's lemma to derive new combinatorial lower bounds for the chromatic number of graphs, enhancing existing topological bounds and providing deeper insights into graph coloring problems.

Contribution

It introduces a generalized form of Gale's lemma and applies it to establish two novel combinatorial lower bounds for graph chromatic numbers.

Findings

Derived new combinatorial bounds for conid(B_0(G))+1 and conid(B(G))+2.

Connected the generalized Gale's lemma to topological bounds on chromatic number.

Enhanced understanding of graph coloring through combinatorial and topological methods.

Abstract

In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two combinatorial sharp lower bounds for ${\rm conid}({\rm B}_0(G))+1$ and ${\rm conid}({\rm B}(G))+2$, two famous topological lower bounds for the chromatic number of a graph $G$.