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3-colored Ramsey Numbers of Odd Cycles
3-colored Ramsey Numbers of Odd Cycles ∗
Annette Schelten, Ingo Schiermeyer
Fakult¨ at f¨ ur Mathematik und Informatik, TU Bergakademie Freiberg, 09596 Freiberg, Germany
Ralph Faudree Dep. of Mathematical Science, University of Memphis, Memphis, TN 38152, USA
Abstract Recently we determined the Ramsey Number r(C7 , C7 , C7 ) = 25.
Let G = (V (G), E(G)) be an undirected finite graph without any loops or multiple edges, where V (G) denotes its vertex set and E(G) its edge set. In the following we will often consider the complete graph Kp on p vertices and the cycle Cp on p vertices. A k−coloring (F1 , F2 , . . . , Fk ) of a graph G is a coloring of the edges of G with at most k different colors F1 , . . . , Fk . The graph < Fi >= (V (G), E(Fi )) denotes the subgraph of G which consists of all vertices of G and all edges which are colored with color Fi . We say that Kp −→ (G1 , G2 , . . . , Gk ), if in each k−coloring of Kp the subgraph < Fi > contains a graph isomorphic to Gi for at least one i with 1 ≤ i ≤ k. Now the Ramsey Number of k graphs G1 , G2 , . . . , Gk is defined as the minimal integer p such that Kp −→ (G1 , . . . , Gk ) (r(G1 , . . . , Gk ) := min{p|Kp −→ (G1 , . . . , Gk )}). A good and detailed overview about known estimations and exact values is given in Radziszowski’s survey ’Small Ramsey Numbers’, [8]. We have a general lower bound only in the case k = 2, namely r(G, H) ≥ (χ(G) − 1)(c(H) − 1) + 1, [3], where χ(G) denotes the chromatic number of G, and c(H) the order of the largest component of H. This estimation is sharp if G is isomorphic to the complete graph on n vertices and H is isomorphic to any tree on m vertices, namely Chvatal proved that r(Kn , Tm ) = (n−1)(m−1)+1 [2]. Preprint submitted to Elsevier Preprint
29 April 1999
In this extended abstact we will mainly concentrate on complete graphs and cycles. The following table (taken from Radziszowski’s survey) contains the results for the classical Ramsey Numbers r(Kk , Kl ) for small k and l. l k 3 4 5 6
3
4
5
6
7
8
9
6
9
14
18
23
28
36
35
49
55
69
18
25
10
11
12
13
14
15
40
46
52
59
66
73
43
51
60
69
78
89
80
96
128
131
136
145
41
61
84
115
149
191
238
291
349
417
43
58
80
95
116
141
153
181
193
221
237
49
87
143
216
316
442
102
109
122
153
167
203
224
242
258
338
165
298
495
780
1171
1031
1713
2826
3583
6090
7 8
205 540
282 1870
565
9
6625
12715 798
10
23854
Due to lack of space we skip the list of all the different authors. This list can also be found in Radziszowski’s survey. Considering two cycles Rosta [7] and independently Faudree und Schelp [5] proved the following general result: 2n − 1 r(Cn , Cm ) = n − 1 + m 2 max{n − 1 +
m 2 , 2m
3 ≤ m ≤ n, m odd, (n, m) = (3, 3) 4 ≤ m ≤ n, m, n even, (n, m) = (4, 4) 4 ≤ m ≤ n, m even n odd
− 1}
In the case k = 3 it becomes even more complicated to find general results or to determine exact Ramsey Numbers. Up to now there is only one known exact value for the so called multicolored (k ≥ 3) classical Ramsey Numbers, namely r(K3 , K3 , K3 ) = r(C3 , C3 , C3 ) = 17 [6]. Considering cycles instead of complete graphs the following Ramsey Numbers are proved: r(C4 , C4 , C4 ) = 11 [1], [4] r(C5 , C5 , C5 ) = 17 [9] and r(C6 , C6 , C6 ) = 12 [10]. These last two numbers are determined by using computer support. Let the complete graph on 4 · (k − 1) vertices be colored as follows: There are 4 subgraphs K 1 , K 2 , K 3 and K 4 , each of order k −1 and completely colored with 2
color F1 . All edges between K 1 and K 2 and all edges between K 3 and K 4 are colored by F2 . Further all the remaining edges are colored with a third color. If k is odd, this coloring contains no monochromatic Ck . Hence we conclude r(Ck , Ck , Ck ) ≥ 4 · (k − 1) + 1 for odd k. The above quoted result shows that this bound is sharp for k = 5. Using a new method we recently proved the following Theorem: Theorem 1 r(C7 , C7 , C7 ) = 25 = 4 · (7 − 1) + 1. Hence the Ramsey Numbers r(Ck , Ck , Ck ) are completely determined for all k ≤ 7. Our method works without any computer support and using analogous arguments we confirm that r(C5 , C5 , C5 ) = 17. Also there is some hope that it helps to determine an upper bound for larger odd k.
References [1] A. Bialostocki and J. Sch¨ onheim, On Some Turan and Ramsey Numbers for C4 , Graph Theory and Combinatorics, Academic Press, London, (1984) 29-33. [2] V. Chvatal, Tree-Complete Graph Ramsey Numbers, Journal of Graph Theory, 1 (1977) 93. [3] V. Chvatal and F. Harary, Generalized Ramsey Theory for Graphs, III. Small Off-Diagonal Numbers, Pacific Journal of Mathematics, 41 (1972) 335-345. [4] C. Clapham, The Ramsey Number r(C4 , C4 , C4 ), Periodica Mathematica Hungarica, 18 (1987) 317-318. [5] R.J. Faudree and R.H. Schelp, All Ramsey Numbers for Cycles in Graphs, Discrete Mathematics, 8 (1974) 313-329. [6] R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal of Mathematics, 7 (1955) 1-7. [7] V. Rosta, On a Ramsey Type Problem of J.A. Bondy and P. Erds, I & II, Journal of Combinatorial Theory, Series B, 15 (1973) 94-120. [8] S.P. Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics 1 (1994), update 1998. [9] Yang Yuangsheng and P. Rowlinson, On the Third Ramsey Numbers of Graphs with Five Edges, Journal of Combinatorial Mathematics and Combinatorial Computing, 11 (1992) 213-222. [10] Yang Yuangsheng and P. Rowlinson, On Graphs without 6-Cycles and Related Ramsey Numbers, Utilitas Mathematica, 44 (1993) 192- 196.
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Annette Schelten, Ingo Schiermeyer
Fakult¨ at f¨ ur Mathematik und Informatik, TU Bergakademie Freiberg, 09596 Freiberg, Germany
Ralph Faudree Dep. of Mathematical Science, University of Memphis, Memphis, TN 38152, USA
Abstract Recently we determined the Ramsey Number r(C7 , C7 , C7 ) = 25.
Let G = (V (G), E(G)) be an undirected finite graph without any loops or multiple edges, where V (G) denotes its vertex set and E(G) its edge set. In the following we will often consider the complete graph Kp on p vertices and the cycle Cp on p vertices. A k−coloring (F1 , F2 , . . . , Fk ) of a graph G is a coloring of the edges of G with at most k different colors F1 , . . . , Fk . The graph < Fi >= (V (G), E(Fi )) denotes the subgraph of G which consists of all vertices of G and all edges which are colored with color Fi . We say that Kp −→ (G1 , G2 , . . . , Gk ), if in each k−coloring of Kp the subgraph < Fi > contains a graph isomorphic to Gi for at least one i with 1 ≤ i ≤ k. Now the Ramsey Number of k graphs G1 , G2 , . . . , Gk is defined as the minimal integer p such that Kp −→ (G1 , . . . , Gk ) (r(G1 , . . . , Gk ) := min{p|Kp −→ (G1 , . . . , Gk )}). A good and detailed overview about known estimations and exact values is given in Radziszowski’s survey ’Small Ramsey Numbers’, [8]. We have a general lower bound only in the case k = 2, namely r(G, H) ≥ (χ(G) − 1)(c(H) − 1) + 1, [3], where χ(G) denotes the chromatic number of G, and c(H) the order of the largest component of H. This estimation is sharp if G is isomorphic to the complete graph on n vertices and H is isomorphic to any tree on m vertices, namely Chvatal proved that r(Kn , Tm ) = (n−1)(m−1)+1 [2]. Preprint submitted to Elsevier Preprint
29 April 1999
In this extended abstact we will mainly concentrate on complete graphs and cycles. The following table (taken from Radziszowski’s survey) contains the results for the classical Ramsey Numbers r(Kk , Kl ) for small k and l. l k 3 4 5 6
3
4
5
6
7
8
9
6
9
14
18
23
28
36
35
49
55
69
18
25
10
11
12
13
14
15
40
46
52
59
66
73
43
51
60
69
78
89
80
96
128
131
136
145
41
61
84
115
149
191
238
291
349
417
43
58
80
95
116
141
153
181
193
221
237
49
87
143
216
316
442
102
109
122
153
167
203
224
242
258
338
165
298
495
780
1171
1031
1713
2826
3583
6090
7 8
205 540
282 1870
565
9
6625
12715 798
10
23854
Due to lack of space we skip the list of all the different authors. This list can also be found in Radziszowski’s survey. Considering two cycles Rosta [7] and independently Faudree und Schelp [5] proved the following general result: 2n − 1 r(Cn , Cm ) = n − 1 + m 2 max{n − 1 +
m 2 , 2m
3 ≤ m ≤ n, m odd, (n, m) = (3, 3) 4 ≤ m ≤ n, m, n even, (n, m) = (4, 4) 4 ≤ m ≤ n, m even n odd
− 1}
In the case k = 3 it becomes even more complicated to find general results or to determine exact Ramsey Numbers. Up to now there is only one known exact value for the so called multicolored (k ≥ 3) classical Ramsey Numbers, namely r(K3 , K3 , K3 ) = r(C3 , C3 , C3 ) = 17 [6]. Considering cycles instead of complete graphs the following Ramsey Numbers are proved: r(C4 , C4 , C4 ) = 11 [1], [4] r(C5 , C5 , C5 ) = 17 [9] and r(C6 , C6 , C6 ) = 12 [10]. These last two numbers are determined by using computer support. Let the complete graph on 4 · (k − 1) vertices be colored as follows: There are 4 subgraphs K 1 , K 2 , K 3 and K 4 , each of order k −1 and completely colored with 2
color F1 . All edges between K 1 and K 2 and all edges between K 3 and K 4 are colored by F2 . Further all the remaining edges are colored with a third color. If k is odd, this coloring contains no monochromatic Ck . Hence we conclude r(Ck , Ck , Ck ) ≥ 4 · (k − 1) + 1 for odd k. The above quoted result shows that this bound is sharp for k = 5. Using a new method we recently proved the following Theorem: Theorem 1 r(C7 , C7 , C7 ) = 25 = 4 · (7 − 1) + 1. Hence the Ramsey Numbers r(Ck , Ck , Ck ) are completely determined for all k ≤ 7. Our method works without any computer support and using analogous arguments we confirm that r(C5 , C5 , C5 ) = 17. Also there is some hope that it helps to determine an upper bound for larger odd k.
References [1] A. Bialostocki and J. Sch¨ onheim, On Some Turan and Ramsey Numbers for C4 , Graph Theory and Combinatorics, Academic Press, London, (1984) 29-33. [2] V. Chvatal, Tree-Complete Graph Ramsey Numbers, Journal of Graph Theory, 1 (1977) 93. [3] V. Chvatal and F. Harary, Generalized Ramsey Theory for Graphs, III. Small Off-Diagonal Numbers, Pacific Journal of Mathematics, 41 (1972) 335-345. [4] C. Clapham, The Ramsey Number r(C4 , C4 , C4 ), Periodica Mathematica Hungarica, 18 (1987) 317-318. [5] R.J. Faudree and R.H. Schelp, All Ramsey Numbers for Cycles in Graphs, Discrete Mathematics, 8 (1974) 313-329. [6] R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal of Mathematics, 7 (1955) 1-7. [7] V. Rosta, On a Ramsey Type Problem of J.A. Bondy and P. Erds, I & II, Journal of Combinatorial Theory, Series B, 15 (1973) 94-120. [8] S.P. Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics 1 (1994), update 1998. [9] Yang Yuangsheng and P. Rowlinson, On the Third Ramsey Numbers of Graphs with Five Edges, Journal of Combinatorial Mathematics and Combinatorial Computing, 11 (1992) 213-222. [10] Yang Yuangsheng and P. Rowlinson, On Graphs without 6-Cycles and Related Ramsey Numbers, Utilitas Mathematica, 44 (1993) 192- 196.
3