DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2022): 0.7

5-year Journal Impact Factor (2022): 0.7

CiteScore (2022): 1.9

SNIP (2022): 0.902

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 25(1-2) (2005) 45-50
DOI: https://doi.org/10.7151/dmgt.1258

PLANAR RAMSEY NUMBERS

Izolda Gorgol

Department of Applied Mathematics
Lublin University of Technology
Nadbystrzycka 38, 20-618 Lublin, Poland
e-mail: I.Gorgol@pollub.pl

Abstract

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.

Keywords: Ramsey number, planar graph, induced subgraph.

2000 Mathematics Subject Classification: 05D10, 05C55.

References

[1] K. Appel and W. Haken, Every planar map is four colourable. Part I. Discharging, Illinois J. Math. 21 (1977) 429-490.
[2] K. Appel, W. Haken, and J. Koch, Every planar map is four colourable. Part II. Reducibility, Illinois J. Math. 21 (1977) 491-567.
[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.
[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.
[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.
[6] I. Gorgol, Planar and induced Ramsey numbers (Ph.D. thesis (in Polish), Adam Mickiewicz University Poznań, Poland, 2000) 51-57.
[7] I. Gorgol and T. Łuczak, On induced Ramsey numbers, Discrete Math. 251 (2002) 87-96, doi: 10.1016/S0012-365X(01)00328-4.
[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.
[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.
[10] B. Grünbaum, Grötzsch's theorem on 3-colorings, Michigan Math. J. 10 (1963) 303-310.
[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.
[12] V. Rödl, A generalization of Ramsey theorem (Ph.D. thesis, Charles University, Prague, Czech Republic, 1973) 211-220.
[13] R. Steinberg and C.A. Tovey, Planar Ramsey number, J. Combin. Theory (B) 59 (1993) 288-296, doi: 10.1006/jctb.1993.1070.
[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.

Received 24 October 2003
Revised 13 January 2005


Close