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On the Crossing Numbers of Cartesian Products of Wheels and Trees

100%
Klešč M., Petrillová J., Valo M.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 2
399-413
EN
Bokal developed an innovative method for finding the crossing numbers of Cartesian product of two arbitrarily large graphs. In this article, the crossing number of the join product of stars and cycles are given. Afterwards, using Bokal’s zip product operation, the crossing numbers of the Cartesian products of the wheel Wn and all trees T with maximum degree at most five are established.
2
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The crossing numbers of join products of paths with graphs of order four

100%
Klešč M., Schrötter S.
Discussiones Mathematicae Graph Theory
|
2011
|
tom 31
|
nr 2
321-331
EN
Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing numbers for join of paths with all graphs of order four, as well as for join of all graphs of order four with n isolated vertices are given.
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On the Crossing Numbers of Cartesian Products of Stars and Graphs of Order Six

86%
Klešč M., Schrötter Š.
Discussiones Mathematicae Graph Theory
|
2013
|
tom 33
|
nr 3
583-597
EN
The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. According to their special structure, the class of Cartesian products of two graphs is one of few graph classes for which some exact values of crossing numbers were obtained. The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. Moreover, except of six graphs, the crossing numbers of Cartesian products G⃞K1,n for all other connected graphs G on five vertices are known. In this paper we are dealing with the Cartesian products of stars with graphs on six vertices. We give the exact values of crossing numbers for some of these graphs and we summarise all known results concerning crossing numbers of these graphs. Moreover, we give the crossing number of G1⃞T for the special graph G1 on six vertices and for any tree T with no vertex of degree two as well as the crossing number of K1,n⃞T for any tree T with maximum degree five.
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