A Note on Graph Characteristics and Hadwiger's Conjecture

TL;DR

This paper explores relationships between graph parameters inspired by topological concepts, providing new bounds on Hadwiger's and chromatic numbers, with implications for Hadwiger's Conjecture.

Contribution

It introduces three novel graph parameters related to the Euler-Poincare characteristic and establishes bounds involving these parameters and classical graph invariants.

Findings

Three graph parameters are shown to be at least as large as those of a complete graph with a certain number of vertices.

Derived upper bounds for Hadwiger and chromatic numbers based on induced subgraph counts.

Applications to Hadwiger's Conjecture are discussed.

Abstract

This is a note on three graph parameters motivated by the Euler-Poincare characteristic for simplicial complex. We show those three graph parameters of a given connected graph $G$ is greater than or equal to that of the complete graph with $\max(h(G),\chi(G))$ vertices. This will yield three different simultaneous upperbounds of both the hadwiger number and chromatic number by means of the number of particular types of induced subgraphs. Some applications to Hadwiger's Conjecture is also discussed.