Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method
نویسندگان
چکیده
The biplanar crossing number cr2(G) of a graph G is min{cr(G1)+ cr(G2)}, where cr is the planar crossing number and G1 ∪ G2 = G. We show that cr2(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) − 2 ≤ Kcr2(G) log2 n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr2(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. We dedicate this paper to our late colleague and friend, Ondrej Sýkora.
منابع مشابه
Biplanar Crossing Numbers I: A Survey of Results and Problems
We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of speci c families of graphs, in particular, of complete bipartite graphs. We nd a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We nd the exact biplanar crossing number of K5;q for every q.
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عنوان ژورنال:
- Random Struct. Algorithms
دوره 33 شماره
صفحات -
تاریخ انتشار 2008