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Discussiones Mathematicae Graph Theory 33(3) (2013) 571-582
DOI: https://doi.org/10.7151/dmgt.1684

The Crossing Numbers of Products of Path with Graphs of Order Six

Marián Klešč and Jana Petrillová Faculty of Electrical Engineering and Informatics Technical University of Košice Letná 9, 042 00 Košice, Slovak Republic

Abstract

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path P_n of length n, the crossing numbers of Cartesian products G □ P_n for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G □ P_n for graphs G of order six are studied. Let H denote the unique tree of order six with two vertices of degree three. The main contribution is that the crossing number of the Cartesian product H □ P_n is 2(n −1). In addition, the crossing numbers of G □ P_n for fourty graphs G on six vertices are collected.

Keywords: graph, drawing, crossing number, Cartesian product, path, tree

2010 Mathematics Subject Classification: 05C10, 05C38.

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Received 30 November 2011
Revised 12 April 2012
Accepted 17 April 2012