On even triangulations of 2-connected embedded graphs
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(n(2)) 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(n(3/2)) 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.
even triangulations plane graph graph coloring high genus graph