$r$-fundamental groups of graphs

TL;DR

This paper introduces the concept of r-fundamental groups of graphs, establishing their relation to r-covering maps and neighborhood complexes, and uses these notions to derive obstructions for graph mappings, especially to odd cycles.

Contribution

It defines r-fundamental groups and r-neighborhood complexes for graphs, extending topological concepts to graph theory and providing new tools to analyze graph mappings.

Findings

r-fundamental groups correspond to r-covering maps in graphs

Obstructions to graph maps to odd cycles are derived from r-fundamental groups

Kneser graph K_{2k+1,k} has no graph maps to C_5

Abstract

In this paper, we introduce the notions of $r$-fundamental groups of graphs, $r$-covering maps, and $r$-neighborhood complexes of graphs for a positive integer $r$. There is a natural correspondence between $r$-covering maps and $r$-fundamental groups as is the case of the covering space theory in topology. We can derive obstructions of the existences of graph maps from $r$-fundamental groups. Especially, $r$-fundamental groups gives deep informations about the existences of graph maps to odd cycles. For example, we prove the Kneser graph $K_{2k+1,k}$ has no graph maps to $C_5$. $r$-neighborhood complexes are natural generalization of neighborhood complexes defined by Lov$\acute{\rm a}$sz. We prove that $(2r)$-fundamental groups gives graph theoretical description of the fundamental groups of $r$-neighborhood complexes.