On Hadwiger's Number of a graph with partial information

TL;DR

This paper explores how partial information about a graph can be used to establish upper bounds on Hadwiger's number, highlighting the role of locality in extending known inequalities.

Contribution

It analyzes the extension of upper bounds from chromatic number to Hadwiger's number based on partial graph information, emphasizing the importance of locality.

Findings

Certain bounds for chromatic number extend to Hadwiger's number with partial info.

Locality of inequalities influences their extendability.

Results identify which bounds can be generalized to Hadwiger's number.

Abstract

We investigate the possibility of proving upper bounds on Hadwiger's number of a graph with partial information, mirroring several known upper bounds for the chromatic number. For each such bound we determine whether the corresponding bound for Hadwiger's number holds. Our results suggest that the ``locality'' of an inequality accounts for the existence of such an extension.