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The crossing number of C m × C n is as conjectured for n ≥ m ( m + 1)
Author(s) -
Glebsky Lev Yu.,
Salazar Gelasio
Publication year - 2004
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.20016
Subject(s) - mathematics , conjecture , combinatorics , crossing number (knot theory) , graph , discrete mathematics , intersection (aeronautics) , engineering , aerospace engineering
It has been long conjectured that the crossing number of C m × C n is ( m −2) n , for all m , n such that n ≥ m ≥ 3. In this paper, it is shown that if n ≥ m ( m + 1) and m ≥ 3, then this conjecture holds. That is, the crossing number of C m × C n is as conjectured for all but finitely many n , for each m . The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by Adamsson and Richter. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 53–72, 2004