NDLI: On Even Triangulations of 2-Connected Embedded Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Zhang, Huaming He, Xin
Copyright Year	2005
Abstract	Recently, Hoffmann and Kriegel proved an important combinatorial theorem [SIAM J. Discrete Math., 9 (1996), pp. 210--224]: Every 2-connected bipartite plane multigraph G without 2-cycle faces has a triangulation in which all vertices have even degree (this is called an even triangulation). Combined with the classical Whitney's theorem, this result implies that every such graph has a 3-colorable plane triangulation. Using this theorem, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. A complicated O(n2 ) time algorithm was obtained in [SIAM J. Discrete Math., 9 (1996), pp. 210--224] for constructing an even triangulation of G. Hoffmann and Kriegel conjectured that there is an O(n3/2 ) time algorithm for solving this problem.In this paper, we develop a simple proof of the above theorem. Our proof reveals and relies on a natural correspondence between even triangulations of G and certain orientations of G. Based on this new proof, we obtain a very simple O(n) time algorithm for finding an even triangulation of G. We also extend our proof to show the existence of even triangulations for similar graphs on high genus surface.
Starting Page	683
Ending Page	696
Page Count	14
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539702408247
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3
Volume Number	34
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-27
Access Restriction	Subscribed
Subject Keyword	plane graph even triangulations Applications Vertex degrees graph coloring Graph theory Graph algorithms high genus graph Coloring of graphs and hypergraphs
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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