2001
DOI: 10.1006/jctb.2000.2015
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Drawings of Cm×Cn with One Disjoint Family II

Abstract: A long-standing conjecture states that the crossing number of the Cartesian product of cycles C m _C n is (m&2) n, for every m, n satisfying n m 3. A crossing is proper if it occurs between edges in different principal cycles. In this paper drawings of C m _C n with the principal n-cycles pairwise disjoint or the principal m-cycles pairwise disjoint are analyzed, and it is proved that every such drawing has at least (m&2) n proper crossings. As an application of this result, we prove that the crossing number o… Show more

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Cited by 7 publications
(5 citation statements)
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“…toroidal grid (a Cartesian product of two cycles) minor in G, and then used known lower bounds [22] on its crossing number to derive our conclusions (cf. Lemma 6.1).…”
Section: A Standalone Lower Boundmentioning
confidence: 99%
“…toroidal grid (a Cartesian product of two cycles) minor in G, and then used known lower bounds [22] on its crossing number to derive our conclusions (cf. Lemma 6.1).…”
Section: A Standalone Lower Boundmentioning
confidence: 99%
“…3) is crðC m  C n Þ ! ð1=2Þðm À 2Þn [10]. Using this bound, we obtain the following slightly improved version of Theorem 1.1.…”
Section: Discussionmentioning
confidence: 87%
“…For n between 5ðm À 1Þ=4 and ðm þ 1Þ ðmþ2Þ=2, the best general lower bound known is crðC m ÂC n Þ ! ðm À 2Þn=2 [10].…”
Section: Discussionmentioning
confidence: 99%
“…It was recently proved by Glebsky and Salazar [6] that the crossing number of C m × C n equals its long-conjectured value at least for n m(m + 1). General lower bounds within a constant multiplicative factor of the conjectured value for cr(C m × C n ) were only proved recently in [10,19]. It is proved in [10] that cr(C m × C n ) 1 9 mn and in [19] that cr(C m × C n ) 1 2 (m − 2)n. The crossing numbers of the Cartesian products of cycles and all graphs of order four are determined in [4,9].…”
Section: Introductionmentioning
confidence: 94%