In this article, we will determine the crossing number of the complete tripartite graphs K,.3.n and K2,3.n. Our proof depends on Kleitman's results for the complete bipartite graphs [D. J. Kleitman, The crossing number of K5,n. J. Combhatorial Theory 9 (1970) 375-3231. a graph G is the minimum numbe
On Integers of the Forms k−2n and k2n+1
✍ Scribed by Yong-Gao Chen
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 85 KB
- Volume
- 89
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
In this paper we prove that the set of positive odd integers k such that k&2 n has at least three distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to k2 n +1 is also true.
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## Abstract For complete __i__‐partite graphs of the form __K(n__~1~, __n, n__, …, __n__) the largest value of __n__~1~ that allows the graph to be triangularly‐embedded into a surface is (__i__‐2)__n.__ In this paper the author constructs triangular embeddings into surfaces of some complete partit
A 1-factor of a graph G = (V, E) is a collection of disjoint edges which contain all the vertices of V . Given a 2n -1 edge coloring of K2n, n ≥ 3, we prove there exists a 1-factor of K2n whose edges have distinct colors. Such a 1-factor is called a ''Rainbow.''
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In this paper we define 2-adic cyclotomic elements in K-theory and e tale cohomology of the integers. We construct a comparison map which sends the 2-adic elements in K-theory onto 2-adic elements in cohomology. Using calculation of 2-adic K-theory of the integers due to Voevodsky, Rognes and Weibel