r-fundamental groups of graphs

2013/01/30 by Takahiro Matsushita, Matsushita, Takahiro
Mathematics · #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #math.AT #math.CO

paper · pdf · doi:10.48550/arxiv.1301.7217

This paper has been withdrawn by the author since he wrote the results of this paper in the latest version of arXiv:1210.2803

arxiv created 2015/09/13 · arxiv updated 2015/09/15

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 C₅. r-neighborhood complexes are natural generalization of neighborhood complexes defined by Lov\acute\rm asz. We prove that (2r)-fundamental groups gives graph theoretical description of the fundamental groups of r-neighborhood complexes.

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