On the crossing numbers of Cartesian products with paths
نویسنده
چکیده
Using a newly introduced operation on graphs and its counterpart on graph drawings, we prove the conjecture of Jendrol’ and Ščerbová from 1982 about the crossing number of the Cartesian product K1,m2Pn. Our approach is applicable to the capped Cartesian products of Pn with any graph containing a dominating vertex.
منابع مشابه
On the crossing numbers of Cartesian products of paths with special graphs
There are known exact results of the crossing numbers of the Cartesian product of all graphs of order at most four with paths, cycles and stars. Moreover, for the path Pn of length n, the crossing numbers of Cartesian products G Pn for all connected graphs G on five vertices and for forty graphs G on six vertices are known. In this paper, we extend these results by determining the crossing numb...
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The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path Pn of length n, the crossing numbers of Cartesian products G Pn for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G Pn for graphs G of order six are studied. Let H denote the unique tree of order si...
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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 ...
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There are several known exact results on the crossing numbers of Cartesian products of paths, cycles or stars with “small” graphs. Let H be the 5-vertex graph defined from K5 by removing three edges incident with a common vertex. In this paper, we extend the earlier results to the Cartesian products of H ×Pn and H ×Cn, showing that in the general case the corresponding crossing numbers are 3n−1...
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ورودعنوان ژورنال:
- J. Comb. Theory, Ser. B
دوره 97 شماره
صفحات -
تاریخ انتشار 2007