Discussiones Mathematicae Graph Theory 32(2) (2012) 289-297
doi: 10.7151/dmgt.1607

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1-factors and Characterization of Reducible Faces of Plane Elementary Bipartite Graphs

Andrej Taranenko and Aleksander Vesel

Faculty of Natural Sciences and Mathematics
University of Maribor
Koroška cesta 160, 2000 Maribor, Slovenia

Abstract

As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekulé structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given.

A peripheral face f of a plane elementary graph is reducible, if the removal of the internal vertices and edges of the path that is the intersection of f and the outer cycle of G results in an elementary graph. We characterize the reducible faces of a plane elementary bipartite graph. This result generalizes the characterization of reducible faces of an elementary benzenoid graph.

Keywords: plane elementary bipartite graph, reducible face, perfect matching, 1-factor, benzenoid graph

2010 Mathematics Subject Classification: 05C70.

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Received 30 November 2010
Revised 11 May 2011
Accepted 24 May 2011