Let Kl×t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For simple graphs G and H, the size multipartite Ramsey number mj (G, H) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kl×t contains a red G or a blue H as a subgraph. In particular, if j = 3 then m3(G, H) is called the size tripartite Ramsey number of G and H. In this paper, we determine the exact values of the size tripartite numbers m3(P3, mK1,n) for all integers m ≥ 1 and n ≥ 3, where P3 is a path of order 3 and mK1,n is a disjoint union of m copies of a star K1,n.

Topics
Ramsey's theory
1.
A. P.
 
Burger
 and
J. H.
 
van Vuuren
,
Ramsey numbers in complete balanced multipartite graphs. Part II: Size Numbers
,
Discrete Math.
 
283
,
45
–
49
(
2004
).
https://doi.org/10.1016/j.disc.2004.02.003
Google Scholar
Crossref
Search ADS
 
2.
D.
 
Day
,
W.
 
Goddard
,
M. A.
 
Henning
 and
H. C.
 
Swart
,
Multipartite Ramsey numbers
,
Ars Combin.
 
58
,
23
–
31
(
2001
).
Google Scholar
 
3.
Syafrizal
 
Sy
,
E. T.
 
Baskoro
, and
S.
 
Uttunggadewa
,
The size multipartite Ramsey numbers for paths
,
J. Combin. Math. Combin.
 
55
,
103
–
107
(
2005
).
Google Scholar
 
4.
Syafrizal
 
Sy
,
E. T.
 
Baskoro
, and
S.
 
Uttunggadewa
,
The size multipartite Ramsey numbers for small paths versus other graphs
,
Far East J. Appl. Math.
 
28
Issue
1
,
131
–
138
(
2007
).
Google Scholar
 
5.
Syafrizal
 
Sy
,
E. T.
 
Baskoro
,
S.
 
Uttunggadewa
 and
H.
 
Assiyatun
,
Path-path size multipartite Ramsey numbers
,
J. Combin. Math. Combin. Comput.
 
71
,
265
–
271
(
2009
).
Google Scholar
 
6.
Syafrizal
 
Sy
,
On size multipartite Ramsey numbers for paths versus cycle of three or four vertices
,
Far East J. Appl. Math.
Volume
44
No.
2
,
109
–
116
(
2010
).
Google Scholar
 
7.
Syafrizal
 
Sy
,
On size multipartite Ramsey numbers for small paths versus cocktail party graphs
,
Far East J. Appl. Math.
Volume
55
No.
1
,
53
–
60
(
2011
).
Google Scholar
 
8.
Syafrizal
 
Sy
 and
E. T.
 
Baskoro
,
Lower bounds for size multipartite Ramsey numbers mj (Pn, Kj×b
),
American Institute of Physics Proceeding Volume 1450
,
259
–
261
(
2011
).
Google Scholar
Crossref
Search ADS
 
9.
Syafrizal
 
Sy
,
The size multipartite Ramsey numbers of paths with respect to complete balanced multipartite graphs
,
Far East J. Appl. Math.
Volume
62
No.
2
,
107
–
115
(
2012
).
Google Scholar
 
10.
Syafrizal
 
Sy
,
Tree-path size bipartite Ramsey numbers
,
Far East J. Math. Sci.
Volume
76
No.
1
,
139
–
145
(
2013
).
Google Scholar
 
11.
Surahmat
 and
Syafrizal
 
Sy
,
Star-path size multipartite Ramsey numbers
,
Applied Math. Sciences (Bulgaria)
Volume
8
No.
75
,
3733
–
3736
(
2014
).
Google Scholar
 
This content is only available via PDF.
© 2016 AIP Publishing LLC.
2016
AIP Publishing LLC
You do not currently have access to this content.