L. W. Beineke and F. Harary, The thickness of the complete graph. Canad. J. Math., 17:850-859, 1965. Home/Search Document Not in Database Summary Related Articles Check |
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Geometric Thickness in a Grid - Wood (2001) (Correct)
....lower and upper bounds for the geometric thickness of complete and complete bipartite graphs. It is shown that d n 1 5:646 e (Kn ) d 4 e. Note that their construction has O(n ) area under the vertex resolution rule [D. Eppstein, personal communication] Since (Kn ) 6 O(1) [3, 4, 6, 21], Kn ) Kn ) for large enough n. On the other hand, K a;b ) K a;b ) when a b [11] Another parameter closely related to geometric thickness is that of book thickness, introduced by Bernhart and Kainen [7] A book consists of a line in 3 space, called the spine, and some number of ....
L. W. Beineke and F. Harary, The thickness of the complete graph. Canad. J. Math., 17:850-859, 1965.
Planarizing Graphs - A Survey and Annotated Bibliography - Liebers (2001) (6 citations) (Correct)
....3 ) #(K 4 ) 1, and it is easily seen that #(K 5 ) #(K 6 ) #(K 7 ) #(K 8 ) 2. Figure 12 shows that #(K 9 ) # 3. Battle, Harary, and Kodama [BHK62] were the first to show that indeed #(K 9 ) 3. Alternative proofs were given by Tutte [Tut63a] and Wessel [Wes86] Beineke and Harary [BH65] showed the formula for #(Kn ) for most cases, and Alekseev and Goncakov [AG76] and, independently, Vasak [Vas76] completed the result: #(Kn ) # n 7 6 # for n # 1, n #= 9, n #= 10 #(K 9 ) #(K 10 ) 3 For the complete bipartite graph, the thickness is still not settled for ....
Lowell W. Beineke and Frank Harary. The thickness of the complete graph. Can. J. Math., 17:850--859, 1965.
The Thickness of Graphs: A Survey - Mutzel, Odenthal, Scharbrodt (1998) (3 citations) (Correct)
....with jV j = n (n 2) and jEj = m, then i) G) d m 3n Gamma6 e, ii) G) d m 2n Gamma4 e, if G has no triangles. 3 In the 1960 s the cornerstone of the thickness work on special graph classes was laid by Harary and Beineke, who published the first results on the thickness of complete [BH65] and complete bipartite graphs [BHM64] But the determination of a nice formula describing the thickness of complete graphs has a long history and was completed by Alekseev and Goncakov [AG76] By the way, the question of whether (K 16 ) 3 or (K 16 ) 4 gives rise to a little anecdote, ....
Beineke, L.W., and F. Harary, The thickness of the complete graph, Canad. J. Math. 17 (1965), 850--859.
Geometric Thickness of Complete Graphs - Dillencourt, Eppstein, Hirschberg (1999) (8 citations) (Correct)
....be reassigned arbitrarily in the plane without altering the topology of the planar embedding provided we are allowed to bend the edges at will [10] This observation is easily verified by induction, moving one vertex at a time. The (graph theoretical) thickness is now known for all complete graphs [1, 2, 3, 12, 13], and is given by the following formula: Kn ) 8 : 1; 1 n 4 2; 5 n 8 3; 9 n 10 Sigma n 2 6 Upsilon ; n 10 (1:1) Another notion related to geometric thickness is the book thickness of a graph G, bt(G) defined as follows [4] A book with k pages or a k book , is a ....
L. W. Beineke and F. Harary. The thickness of the complete graph. Canadian Journal of Mathematics, 17:850--859, 1965.
The Visibility Number Of A Graph - Chang, Jacobson, Lehel, West (1998) (Correct)
....the visibility number of an n vertex graph G, it is tempting to use Remark 1 and express G as a union of planar graphs, since planar graphs have visibility at 7 most 2. When n is not 9 or 10, every n vertex graph is the union of at most b(n 7) 6c planar graphs, with equality for cliques (see [1] for n 6j 4 mod 6, with the remaining case settled through the work of many authors) This yields about n=3 as an upper bound on visibility number of n vertex graphs. Using planar graphs whose cut vertices lie on one face, which seems feasible, could eliminate the factor of 2 between this and the ....
L.W. Beineke and F. Harary, The thickness of the complete graph. Canad. J. Math. 17 (1965), 464--496.
Remarks On The Thickness Of A Graph - Aho, Mäkinen, Systä (1996) (1 citation) (Correct)
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L.W. Beineke and F. Harary, The thickness of the complete graph. Canadian J. Math. 17:850-859 (1965).
Remarks On The Thickness Of A Graph - Aho, Mäkinen, Systä (1996) (1 citation) (Correct)
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L.W. Beineke and F. Harary, The thickness of the complete graph. ########### ##### 17:850-859 (1965).
Geometric Thickness of Complete Graphs - Dillencourt, Eppstein, Hirschberg (2000) (8 citations) (Correct)
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L. W. Beineke and F. Harary. The thickness of the complete graph. Canadian Journal of Mathematics, 17:850--859, 1965.
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