Geodesic
A Note on the Ramsey Number of Even Wheels Versus Stars
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 397-404.

Voir la notice de l'article provenant de la source Library of Science

For two graphs G_1 and G_2, the Ramsey number R(G_1,G_2) is the smallest integer N, such that for any graph on N vertices, either G contains G_1 or G contains G_2. Let S_n be a star of order n and W_m be a wheel of order m + 1. In this paper, we will show R(W_n, S_n) ≤ 5n//2 − 1, where n ≥ 6 is even. Also, by using this theorem, we conclude that R(W_n, S_n) = 5n//2 − 2 or 5n//2 −1, for n ≥ 6 and even. Finally, we prove that for sufficiently large even n we have R(W_n, S_n) = 5n//2 − 2.
Keywords: Ramsey number, star, wheel, weakly pancyclic 
@article{DMGT_2018_38_2_a4,
     author = {Haghi, Sh. and Maimani, H.R.},
     title = {A {Note} on the {Ramsey} {Number} of {Even} {Wheels} {Versus} {Stars}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {397--404},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a4/}
}
TY  - JOUR
AU  - Haghi, Sh.
AU  - Maimani, H.R.
TI  - A Note on the Ramsey Number of Even Wheels Versus Stars
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2018
SP  - 397
EP  - 404
VL  - 38
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a4/
LA  - en
ID  - DMGT_2018_38_2_a4
ER  - 
%0 Journal Article
%A Haghi, Sh.
%A Maimani, H.R.
%T A Note on the Ramsey Number of Even Wheels Versus Stars
%J Discussiones Mathematicae. Graph Theory
%D 2018
%P 397-404
%V 38
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a4/
%G en
%F DMGT_2018_38_2_a4
Haghi, Sh.; Maimani, H.R. A Note on the Ramsey Number of Even Wheels Versus Stars. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 397-404. http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a4/

[1] S. Brandt, R. Faudree and W. Goddard, Weakly pancyclic graphs, J. Graph Theory 27 (1998) 141–176. doi:10.1002/(SICI)1097-0118(199803)27:3h141::AID-JGT3i3.0.CO;2-O

[2] G. Chartrand and O.R. Oellermann, Applied and Algorithmic Graph Theory (Mc Graw-Hill Inc, 1993).

[3] Y. Chen, Y. Zhang and K. Zhang, The Ramsey numbers of stars versus wheels, European J. Combin. 25 (2004) 1067–1075. doi:10.1016/j.ejc.2003.12.004

[4] V. Chvátal and F. Harary, Generalized Ramsey theory for graphs, III. Small offdiagonal numbers, Pac. J. Math. 41 (1972) 335–345. doi:10.2140/pjm.1972.41.335

[5] G.A. Dirac, Some theorems on abstract graphs, Proc. Lond. Math. Soc. 2 (1952) 69–81. doi:10.1112/plms/s3-2.1.69

[6] R.J. Faudree and R.H. Schelp, All Ramsey numbers for cycles in graphs, Discrete Math. 8 (1974) 313–329. doi:10.1016/0012-365X(74)90151-4

[7] J.M. Hasmawati, Bilangan Ramsey untuk Graf Bintang Terhadap Graf Roda (Tesis Magister, Departemen Matematika ITB, Indonesia, 2004).

[8] E.T. Hasmawati, E.T. Baskoro and H. Assiyatun, Star-wheel Ramsey numbers, J. Combin. Math. Combin. Comput. 55 (2005) 123–128.

[9] A. Korolova, Ramsey numbers of stars versus wheels of similar sizes, Discrete Math. 292 (2005) 107–117. doi:10.1016/j.disc.2004.12.003

[10] B. Li and I. Schiermeyer, On star-wheel Ramsey numbers, Graphs Combin. 32 (2016) 733–739. doi:10.1007/s00373-015-1594-6

[11] V. Rosta, On a Ramsey-type problem of J.A. Bondy and P. Erdös, II, J. Combin. Theory Ser. B 15 (1973) 105–120. doi:10.1016/0095-8956(73)90036-1

[12] Surahmat and E.T. Baskoro, On the Ramsey number of a path or a star versus W 4 or W 5, in: Proceedings of the 12th Australasian Workshop on Combinatorial Algorithms (Bandung, Indonesia, July, 2001) 174–179.

[13] Y. Zhang, T.C.E. Cheng and Y. Chen, The Ramsey numbers for stars of odd order versus a wheel of order nine, Discrete Math. Algorithms Appl. 1 (2009) 413–436. doi:10.1142/S1793830909000336

[14] Y. Zhang, Y. Chen and K. Zhang, The Ramsey numbers for stars of even order versus a wheel of order nine, European J. Combin. 29 (2008) 1744–1754. doi:10.1016/j.ejc.2007.07.005