The planar Ramsey number PR(G,H) is defined as the smallest integer n for which any 2-colouring of edges of Kₙ with red and blue, where red edges induce a planar graph, leads to either a red copy of G, or a blue H. In this note we study the weak induced version of the planar Ramsey number in the case when the second graph is complete.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1258, author = {Izolda Gorgol}, title = {Planar Ramsey numbers}, journal = {Discussiones Mathematicae Graph Theory}, volume = {25}, year = {2005}, pages = {45-50}, zbl = {1077.05063}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1258} }
Izolda Gorgol. Planar Ramsey numbers. Discussiones Mathematicae Graph Theory, Tome 25 (2005) pp. 45-50. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1258/
[000] [1] K. Appel and W. Haken, Every planar map is four colourable. Part I. Discharging, Illinois J. Math. 21 (1977) 429-490. | Zbl 0387.05009
[001] [2] K. Appel, W. Haken, and J. Koch, Every planar map is four colourable. Part II. Reducibility, Illinois J. Math. 21 (1977) 491-567. | Zbl 0387.05010
[002] [3] W. Deuber, A generalization of Ramsey's theorem, in: R. Rado, A. Hajnal and V. Sós, eds., Infinite and finite sets, vol. 10 (North-Holland, 1975) 323-332.
[003] [4] P. Erdős, A. Hajnal and L. Pósa, Strong embeddings of graphs into colored graphs, in: R. Rado, A. Hajnal and V. Sós, eds., Infinite and finite sets, vol. 10 (North-Holland, 1975) 585-595. | Zbl 0312.05123
[004] [5] I. Gorgol, A note on a triangle-free - complete graph induced Ramsey number, Discrete Math. 235 (2001) 159-163, doi: 10.1016/S0012-365X(00)00269-7. | Zbl 0978.05053
[005] [6] I. Gorgol, Planar and induced Ramsey numbers (Ph.D. thesis (in Polish), Adam Mickiewicz University Poznań, Poland, 2000) 51-57.
[006] [7] I. Gorgol and T. Łuczak, On induced Ramsey numbers, Discrete Math. 251 (2002) 87-96, doi: 10.1016/S0012-365X(01)00328-4. | Zbl 1004.05042
[007] [8] R.E. Greenwood and A.M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math. 7 (1955) 1-7, doi: 10.4153/CJM-1955-001-4. | Zbl 0064.17901
[008] [9] H. Grötzsch, Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math. Natur. Reihe 8 (1958/1959) 109-120.
[009] [10] B. Grünbaum, Grötzsch's theorem on 3-colorings, Michigan Math. J. 10 (1963) 303-310. | Zbl 0115.40903
[010] [11] N. Robertson, D. Sanders, P.D. Seymour and R. Thomas, The four-colour theorem, J. Combin. Theory (B) 70 (1997) 145-161, doi: 10.1006/jctb.1997.1750. | Zbl 0883.05056
[011] [12] V. Rödl, A generalization of Ramsey theorem (Ph.D. thesis, Charles University, Prague, Czech Republic, 1973) 211-220.
[012] [13] R. Steinberg and C.A. Tovey, Planar Ramsey number, J. Combin. Theory (B) 59 (1993) 288-296, doi: 10.1006/jctb.1993.1070. | Zbl 0794.05091
[013] [14] K. Walker, The analog of Ramsey numbers for planar graphs, Bull. London Math. Soc. 1 (1969) 187-190, doi: 10.1112/blms/1.2.187. | Zbl 0184.27705