The crossing numbers of join products of paths with graphs of order four

• Autorzy: Marián Klešč , Stefan Schrötter
• Języki publikacji: EN

Abstrakty

EN

Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing numbers for join of paths with all graphs of order four, as well as for join of all graphs of order four with n isolated vertices are given.

Słowa kluczowe

EN

graph; drawing; path; crossing number; join product

Kategorie tematyczne

• 05C10: Planar graphs; geometric and topological aspects of graph theory
• 05C38: Paths and cycles

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2011
• Tom: 31
• Numer: 2
• Strony: 321-331

Daty

wydano

2011

otrzymano

2009-11-30

poprawiono

2010-06-22

zaakceptowano

2010-06-28

Twórcy

autor

Marián Klešč

• Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University, 042 00 Košice, Slovak Republic

autor

Stefan Schrötter

• Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University, 042 00 Košice, Slovak Republic

Bibliografia

• [1] K. Asano, The crossing number of $K_{1,3,n}$ and $K_{2,3,n}$, J. Graph Theory 10 (1986) 1-8, doi: 10.1002/jgt.3190100102.
• [2] L.W. Beineke and R.D. Ringeisen, On the crossing numbers of products of cycles and graphs of order four, J. Graph Theory 4 (1980) 145-155, doi: 10.1002/jgt.3190040203.
• [3] D. Bokal, On the crossing numbers of Cartesian products with paths, J. Combin. Theory (B) 97 (2007) 381-384, doi: 10.1016/j.jctb.2006.06.003.
• [4] M.R. Garey and D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic. Discrete Methods 4 (1983) 312-316, doi: 10.1137/0604033.
• [5] L.Y. Glebsky and G. Salazar, The crossing number of Cₘ ×Cₙ is as conjectured for n ≥; m(m+1), J. Graph Treory 47 (2004) 53-72, doi: 10.1002/jgt.20016.
• [6] F. Harary, P.C. Kainen and A.J. Schwenk, Toroidal graphs with arbitrarily high crossing numbers, Nanta Math. 6 (1973) 58-67.
• [7] S. Jendrol' and M. Scerbová, On the crossing numbers of Sₘ ×Pₙ and Sₘ ×Cₙ, Casopis pro pestování matematiky 107 (1982) 225-230.
• [8] M. Klešč, The crossing numbers of Cartesian products of paths with 5-vertex graphs, Discrete Math. 233 (2001) 353-359, doi: 10.1016/S0012-365X(00)00251-X.
• [9] M. Klešč, The join of graphs and crossing numbers, Electronic Notes in Discrete Math. 28 (2007) 349-355, doi: 10.1016/j.endm.2007.01.049.
• [10] D.J. Kleitman, The crossing number of $K_{5,n}$, J. Combin. Theory (B) 9 (1971) 315-323.
• [11] V.R. Kulli and M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97.
• [12] K. Zarankiewicz, On a problem of P. Turán concerning graphs, Fund. Math. 41 (1954) 137-145.

Identyfikatory

DOI

10.7151/dmgt.1548