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The Crossing Number of The Hexagonal Graph H 3,n

R eferences [1] J. Adamsson and R.B. Richter, Arrangements, circular arrangements and the crossing number of C 7 × C n , J. Combin. Theory Ser. B 90 (2004) 21–39. doi:10.1016/j.jctb.2003.05.001 [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 Ser. B 97 (2007) 381–384. doi:10.1016/j.jctb.2006.06.003 [4] D. Bokal, On the

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The Crossing Number of Join of the Generalized Petersen Graph P(3, 1) with Path and Cycle

R eferences [1] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan Press Ltd, London, 1976). [2] P. Erdős and R.K. Guy, Crossing number problems , Amer. Math. Monthly 80 (1973) 52–58. doi:10.2307/2319261 [3] 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 [4] V.R. Kulli and M.H. Muddebihal, Characterization of join graphs with crossing number zero , Far East J. Appl. Math. 5 (2001) 87–97. [5] D.J. Kleitman, The

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The Crossing Numbers of Products of Path with Graphs of Order Six

References [1] 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 [2] D. Bokal, On the crossing number of Cartesian products with paths, J. Combin. Theory (B) 97 (2007) 381-384. doi:10.1016/j.jctb.2006.06.003 [3] S. Jendrol’ and M. Ščerbová, On the crossing numbers of Sm × Pn and Sm × Cn, ˇ Casopis Pro P ˇ estov´ an´ı Matematiky 107 ( 1982) 225-230. [4] M. Klešč, The

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On the Crossing Numbers of Cartesian Products of Stars and Graphs of Order Six

References 1] K. Asano, The crossing number of K1,3,n and K2,3,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 number of Cartesian products with paths, J. Combin. Theory (B) 97 (2007) 381-384. doi:10.1016/j.jctb.2006.06.003 [4] D. Bokal, On the crossing numbers of Cartesian

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On the Crossing Numbers of Cartesian Products of Wheels and Trees

References [1] D. Archdeacon and R.B. Richter, On the parity of crossing numbers, J. Graph Theory 12 (1988) 307-310. doi: 10.1002/jgt.3190120302 [2] K. Asano, The crossing number of K1,3,n and K2,3,n, J. Graph Theory 10 (1986) 1-8. doi: 10.1002/jgt.3190100102 [3] 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 [4] D. Bokal, On the crossing numbers of Cartesian products with

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A Note on the Crossing Numbers of 5-Regular Graphs

crossing number of join of the generalized Petersen graph P (3, 1) with path and cycle , Discuss. Math. Graph Theory 38 (2018) 351–370. doi:10.7151/dmgt.2005 [5] M. Schaefer, Crossing Numbers of Graphs (CRC Press Inc., Boca Raton, Florida, 2017). [6] Y.S. Yang, J.H. Lin and Y.J. Dai, Largest planar graphs and largest maximal planar graphs of diameter two , J. Comput. Appl. Math. 144 (2002) 349–358. doi:10.1016/S0377-0427(01)00572-6

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The Crossing Numbers of Join of Some Graphs with n Isolated Vertices

R eferences [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] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications (North-Holland, New York-Amsterdam-Oxford, 1982). [3] P. Erdős and R.K. Guy, Crossing number problems , Amer. Math. Monthly 80 (1973) 52–58. doi:10.2307/2319261 [4] P.T. Ho, On the crossing number of K 1 ,m,n , Discrete Math. 308 (2008) 5996–6002. doi:10.1016/j.disc.2007.11.023 [5] Y. Huang and T. Zhao, The crossing number of K 1

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2-COLORED ARCHETYPAL PERMUTATIONS AND STRINGS OF DEGREEn

Abstract

New notions related to permutations are introduced here. We define the string of a 2-colored permutation as a closed planar curve, the fundamental 2- colored permutation as an equivalence class related to the equivalence in strings of the 2-colored permutations. We establish an algorithm to identify the 2-colored archetypal permutations of degree n. We present a formula for the number of the 2-colored archetypal permutations of degree n. We describe all the closed planar curves with crossing number ≤ 2 using the 2-colored archetypal permutations. We also present the atlas of the 2- colored archetypal strings of degree n; n ≤ 5.

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On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

(Springer, New York, 2010). [5] I. Fabrici and T. Madaras, The structure of 1- planar graphs , Discrete Math. 307 (2007) 854–865. doi:10.1016/j.disc.2005.11.056 [6] D.V. Karpov, An upper bound on the number of edges in an almost planar bipartite graph , J. Math. Sci. 196 (2014) 737–746. doi:10.1007/s10958-014-1690-9 [7] J. Pach and G. Tóth, Graphs drawn with few crossings per edge , Combinatorica 17 (1997) 427–439. doi:10.1007/BF01215922 [8] H. Bodendiek, R. Schumacher and K. Wagner, Über 1- optimale Graphen , Math. Nachr. 117 (1984

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On Some Characterizations of Antipodal Partial Cubes

, b ) -partitions , Ars Combin. 51 (1999) 113–119. [14] A. Ilíc, S. Klavžar and M. Milanovíc, On distance-balanced graphs , European J. Combin. 31 (2010) 732–737. doi:10.1016/j.ejc.2009.10.006 [15] J. Jerebic, S. Klavžar and D.F. Rall, Distance-balanced graphs , Ann. Comb. 12 (2008) 71–79. doi:10.1007/s00026-008-0337-2 [16] S. Klavžar and M. Kovše, On even and harmonic-even partial cubes , Ars Combin. 93 (2009) 77–86. [17] S. Klavžar and H.M. Mulder, Partial cubes and crossing graphs , SIAM J. Discrete Math. 15 (2002)) 235

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