NDLI: Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces

Content Provider	Springer Nature Link
Author	Chen, Guantao Chen, Beifang
Copyright Year	2008
Abstract	Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature $$\Phi_G(x) = 1 - \frac{d(x)}{2} + \sum_{\sigma \in F(x)} \frac{1}{\|\sigma\|}$$ as that of [8], where d(x) is the degree of x, F(x) is the multiset of all open faces σ in the embedding such that the closure $$\bar\sigma$$ contains x (the multiplicity of σ is the number of times that x is visited along ∂σ), and \|σ\| is the number of sides of edges bounding the face σ. In this paper, we first show that if the absolute total curvature ∑ x∈V(G) \|Φ G (x)\| is finite, then G has only finite number of vertices of non-vanishing curvature. Next we present a Gauss-Bonnet formula for embedded infinite graphs with finite number of accumulation points. At last, for a finite simple graph G with 3 ≤ d G (x) < ∞ and Φ G (x) > 0 for every x ∈ V(G), we have (i) if G is embedded in a projective plane and #(V(G)) = n ≥ 1722, then G is isomorphic to the projective wheel P n ; (ii) if G is embedded in a sphere and #(V(G)) = n ≥ 3444, then G is isomorphic to the sphere annulus either A n or B n ; and (iii) if d G (x) = 5 for all x ∈ V(G), then there are only 49 possible embedded plane graphs and 16 possible embedded projective plane graphs.
Ending Page	183
Page Count	25
Starting Page	159
File Format	PDF
ISSN	09110119
e-ISSN	14355914
Journal	Graphs and Combinatorics
Issue Number	3
Volume Number	24
Language	English
Publisher	Springer Japan
Publisher Date	2008-06-12
Publisher Place	Japan
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Enumeration in graph theory Topology of $E^2$ , $2$ -manifolds Gauss-Bonnet formula Embedding Finiteness theorem Engineering Design Asymptotic enumeration Relations with graph theory Face cycle Planar graphs; geometric and topological aspects of graph theory Combinatorial curvature Euler relation Infinite graph Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science

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