Let F, G, and H be simple graphs. We say F → (G, H) if for every 2-coloring of the edges of F there exist a monochromatic G or H in F. The Ramsey number r(G, H) is defined as min {|V (F)| : F → (G, H)}, the size Ramsey number r̂(G, H) is defined as min {|E(F)| : F → (G, H)}, and the restricted size Ramsey number r*(G, H) is defined as min {|E(F)| : F → (G, H), |V (F)| = r(G, H)}. In this paper we give a lower bound for the restricted size Ramsey number for a path P3 versus Pn. We also give the upper bound and the exact restricted size Ramsey number for some small values of n.
Topics
Ramsey's theory
REFERENCES
1.
E.T.
Baskoro
, Y.
Nuraeni
, and A.A.G.
Ngurah
, Upper bounds for the size Ramsey numbers for P3 versus or Pn
, Journal of Prime Research in Mathematics
2
, 141
–146
(2006
).2.
J.
Beck
, On size Ramsey number of paths, trees, and circuits II, in: Mathematics of Ramsey theory, Algorithms Combin
. 5
(eds. J.
Nesětřil
, V.
Rödl
), Springer
, Berlin
, 1990
, 34
–45
.3.
S.A.
Burr
, A survey of noncomplete Ramsey theory for graphs
, Ann. New York Acad. Sci.
328
, 58
–75
(1979
).4.
P.
Erdös
and R.J.
Faudree
, Size Ramsey functions, in: Sets, graphs, and numbers, Colloq. Math. Soc. Janos Bolyai
60
(eds. G.
Halasz
, et al.), North-Holland Publishing Co.
, Amsterdam
1992
, 219
–238
.5.
P.
Erdös
, R.J.
Faudree
, C.C.
Rousseau
, and R.H.
Schelp
, The size Ramsey number
, Periodica Mathematica Hungarica
, 9
, Issue 1-2
, 145
–161
(1978
).6.
R.J.
Faudree
and R.H.
Schelp
, A survey of results on the size Ramsey number, Paul Erdös and his mathematics, II
(Budapest
, 2002
) 291
–309
.7.
R.J.
Faudree
and J.
Sheehan
, Size Ramsey numbers for small-order graphs
, J. Graph Theory
7
, 53
–55
(1983
).8.
R.J.
Faudree
and J.
Sheehan
, Size Ramsey numbers involving stars
, Discrete Math
46
, 151
–157
(1983
).9.
L.
Gerencsér
and A.
Gyárfás
, On Ramsey-type problems
. Ann. Univ. Eötvös Sect. Math.
10
, 167
–170
(1967
).10.
C.
Imron
and E.T.
Baskoro
, Bilangan Ramsey sisi dari (r̂)(P3, Pn
), Berkala MIPA
16
No. 2
, 7
–12
(2006
).11.
R.
Lortz
and I.
Mengersen
, Size Ramsey results for paths versus stars
, Australas. J. Combin.
, 18
, 3
–12
(1998
).12.
D.R.
Silaban
, E.T.
Baskoro
, and S.
Uttunggadewa
, On the restricted size Ramsey number for P3 versus Cn
, accepted in Procedia Computer Science
.
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