Lov\'{a}sz' original lower bound: Getting tighter bounds and Reducing computational complexity

TL;DR

This paper explores conditions to improve Lovász's original graph bound by increasing neighborhood complex connectivity, simplifies hom complex computations, and discusses the limitations of double mapping cylinders in graph homotopy theory.

Contribution

It provides new conditions for tightening Lovász's bound and simplifies hom complex calculations using topological and categorical insights.

Findings

Conditions for improved Lovász bound based on neighborhood connectivity

Hom complex computation simplified via homotopy pushouts

Limitations of double mapping cylinders in graph homotopy category

Abstract

In this article, we give conditions on a graph under which the Lov\'{a}sz' original bound of the graph can be improved by increasing the topological connectivity of its neighbourhood complex. We also work out conditions under which computing the topological connectivity of hom complex of a pair of graphs can be simplified. In particular, hom complex as a covariant functor acting on a double mapping cylinder of graphs is a homotopy pushout of hom complex functor applied to its subgraphs. We give applications of this result where the computation of hom complexes is simplified. Finally, we explain why double mapping cylinder of graphs does not give a satisfactory definition of homotopy pushout in the category of graphs.