A Note on Altermatic Number

TL;DR

This paper provides a purely combinatorial proof that the kth altermatic number of a graph serves as a tight lower bound for its chromatic number, building on Tucker's lemma.

Contribution

It offers a new combinatorial proof for the relationship between the altermatic number and the chromatic number of graphs.

Findings

Altermatic number is a tight lower bound for chromatic number.

Provides a combinatorial proof of a known bound.

Strengthens the theoretical understanding of graph coloring.

Abstract

In view of Tucker's lemma (an equivalent combinatorial version of the Borsuk- Ulam theorem), the present authors (2013) introduced the kth altermatic number of a graph G as a tight lower bound for the chromatic number of G. In this note, we present a purely combinatorial proof for this result.