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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 Hexagonal Graph H 3 ,N in the Projective Plane

References [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] 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] L.Y. Glebsky and G. Salazar

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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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Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs

R eferences [1] M.O. Albertson, Chromatic number, independence ratio, and crossing number , Ars Math. Contemp. 1 (2008) 1–6. [2] N. Alon, Combinatorial Nullstellensatz , Combin. Probab. Comput. 8 (1999) 7–29. doi:10.1017/S0963548398003411 [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (North-Holland, New York-Amsterdam-Oxford, 1982). [4] L. Ding, G. Wang and G. Yan, Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz , Sci. China Math. 57 (2014) 1875–1882. doi:10.1007/s11425

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Physical Performance During Soccer-7 Competition and Small-Sided Games in U12 Players

possible, with a limit of three ball-contacts per player (3TOU), with no limit of ball-contacts (MAN), with a greater number of players as internal-offensive wildcard players (2WI) or external-offensive wildcard players (4WE). Two SSGs were played with pitch orientation-delimitation, where the aim of the match was to score goals, either crossing the rival goal-line while dribbling the ball without goalkeepers (INV) or by usual scoring techniques while using official goalkeepers (GKP). The technical staff of the team was always present, motivating the players to achieve

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