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From Steiner Triple Systems to 3-sun systems

From Steiner Triple Systems to 3-sun systems. Chin-Mei Fu ( 高金美 ) Tamkang University ( 淡江大學 ). Join work with Y.-L. Lin( 林遠隆 ), N.-H. Jhuang ( 莊柟樺 ), H.-M. Song ( 宋曉明 ). Outline. Steiner triple system 3-sun systems Embed a cyclic Steiner triple system into a 3-sun system.

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From Steiner Triple Systems to 3-sun systems

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  1. From Steiner Triple Systems to 3-sun systems Chin-Mei Fu (高金美) Tamkang University (淡江大學) Join work with Y.-L. Lin(林遠隆), N.-H. Jhuang (莊柟樺),H.-M. Song (宋曉明) 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  2. Outline • Steiner triple system • 3-sun systems • Embed a cyclic Steiner triple system into a 3-sun system 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  3. Steiner triple system • A Steiner triple system (of order n) STS(n) is a 2-(n,3,1) design, • A collection of 3-subsets of an n-set such that any pair of elements of the n-set is contained in a unique one among these 3-sets. • As was shown by Kirkman, a Steiner triple system of order n exists if and only if either n = 0, 1, or n congruent to 1 or 3 (mod 6). 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  4. STS(n) • Examples of Steiner triple systems of small orders are • (1) S1={{1,2,3}} • (2) S2={{1,2,3},{1,4,5},{1,6,7}, {2,4,6},{2,5,7},{3,4,7},{3,5,6}} • (3) S3={{1,2,3}, {4,5,6}, {7,8,9}, {1,4,7}, {2,5,8}, {3,6,9}, {1,5,9}, {2,6,7}, {3,4,8}, {1,6,8}, {2,4,9}, {3,4,6}} 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  5. 1 2 7 4 6 3 5 STS(n) • STS(7) STS(9) S2={{1,2,3},{1,4,5},{1,6,7}, {2,4,6},{2,5,7},{3,4,7},{3,5,6}} 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  6. Decomposition Let G be a simple graph. A decompositionD ofG is acollection ofedge-disjoint subgraphs H1, H2, …, Hm of G such that every edge of G belongs to exactly one Hj for j = 1,2,3,…,m. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  7. T-decomposition • If each member of D is isomorphic to a graph T, then D is calleda T-decompositionof G. • A T-decompositionof G is also called a (G,T)-design. • A STS(n) corresponds to a C3-decompositionof Knor 3-cycle system of order n. • A STS(n) is a (Kn,C3)-design or a (Kn,K3)-design. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  8. Cn N-Sun graph Let Cn be an n-cycle (v1, v2, v3,…, vn). Add n pendent edges v1w1, v2w2, v3w3,…, vnwn to Cn. The resulting graph on 2n vertices is called an n-sun graph, denoted byS(Cn) = [(v1, v2, v3,…, vn), w1, w2, w3,…, wn] . S(Cn) 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  9. Motivation • In 2008, Anitha and Lekshmi proved that • If k is odd, then K2k can be decomposed into k – 1 k-sun graphs and a perfect matching. • If k is even, then K2k can be decomposed into k – 2 k-sun graphs, a perfect matching and a Hamilton cyce. Kk Kk Kk,k 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  10. v4 C3 v1 v2 v3 v5 v6 3-sun graph A 3-sun graph S(C3) contains a 3-cycle (v1, v2, v3) and a 3-matching {v1v4, v2v5, v3v6}, denoted by [(v1, v2, v3), v4, v5, v6]. S(C3) 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  11. What is the value of n such that Kn can be decomposed into 3-sun graphs? i.e. What is the value of n such that(Kn,S(C3))-design exists? 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  12. Necessary condition • If there exists a 3-sun system of order n, then n 0, 1, 4, 9 (mod 12). Sufficient condition? In 1988, Jian-Xing Yin and Bu-Sheng Gong, Existence of G-designs with |V(G)| =6. Combinatorial designs and applications, Huangshan, 201--218, Lecture Notes in Pure and Appl. Math., 126, Dekker, New York, 1990. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  13. When can we get a cyclic S(C3) system of order n? 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  14. Cyclic design • For a cyclic (Kn,G)-design , the vertex set V of Kn can be identified with Zn. That is, the automorphism  can be represented by  = (0, 1, , n1), i.e.  : ii + 1 (mod n) for iZn. • A cyclic (K7,C3)-design:{ {0,1,3}, {1,2,4},{2,3,5},{3,4,6},{4,5,0},{5,6,1},{6,0,2}} 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  15. 1-rotational design (-cyclic) • For a 1-rotational (Kn,G)-design , the vertex set V of Kn can be identified with Zn1 {}. That is, the automorphism  can be represented by  = () (0, 1, , n2) i.e.,  :   , : ii + 1 (mod n1) for i  Zn1. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  16. Difference set • Let {a,b} be an edge in Kv with V= Zv, the difference of the edge {a, b} is denoted by |a – b|, and the difference of the edge {, b} is denoted by . • Let G be a subgraph of Kn.The difference set of G is defined as (G) = {|v – w| |vV(G), wNG(v) –{}} 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  17. 3-cycle system of order v • There exists a STS(v) v  1, 3 (mod 6) • There exists a cyclic STS(v)  v  1, 3 (mod 6) except v = 9. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  18. 0 1 12 2 11 3 10 4 5 9 6 8 7 n 1 (mod 12) • n = 13 • difference set (K13)={1,  2, ...,  6} • 3-sun graphs : • [(1, 2, 4), 5, 7, 10] • [(2, 3, 5), 6, 8, 11] • [(3, 4, 6), 7, 9, 12] • [(4, 5, 7), 8, 10, 0] • … 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  19. Known Results • In 2008, Wu and Lu proved that • For any positive integers k 3 and m,there exists a cyclic (K2(k+m)+1, mCk)-design. • If k is even, then there exists a cyclic (K2p(k+m)+1, mCk)-design for any positive p. • mCk is a graph obtained from Ck by adding m 1 distinct pendent edges to the vertices of Ck. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  20. n=25 • constructed as follows. • Base 3-sun graphs: • [(0, 1, 12); 2, 8, 21] and • [(0, 3, 8); 4, 9,18]. • difference set (K25)={1,2,..., 12} • We have a cyclic3-sun system of order 25 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  21. n  1 (mod 24) • Base 3-sun graphs: • [(0, 1, 6k); k, 4k, 11k-1], [(0, 2k-1,-1+9k/2);2k,5k-1,-1+19k/2], • and • [(0,2j,3k+j);2j+1,5k+j-1,8k+2j], [(0,k+2j-1,7k/2+j-1);k+2j, 11k/2+j-2,9k+2j-2], for j=1, 2, …, k/2-1, • From each $3$-sun we can get two difference triples, these difference triples form a Skolem difference triple system of order 2k. • Therefore, we have a cyclic 3-sun system of order n, n  1 (mod 24). 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  22. n  37 • Base 3-sun graphs: • [(0,1,5);9,12,17],[(0,2,8);13,16,23], • [(0,3,10);16,20,28], • We have a cyclic3-sun system of order 37 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  23. n  13 (mod 24) • Base 3-sun graphs: • [(0,2j,3k+j+1);2j+1,5k+j-1,8k+2j+1], where j=1, 2, …, (k-1)/2. • [(0,k+2j-1,(7k+1)/2+j);k+2j+2,(11k-3)/2+j,9k+2j+2], where j=1, 2, …, (k-5)/2. • [(0,2k-1,5k);2k-4,4k+2,9k-1], [(0,k+2,6k+1);2k-2,3k+4,10k+ 1], [(0,1,(11k+3)/2);2k,2k+2,(19k+5)/2]. • difference triples form an O'Keefe difference triple system of order 2k when k is odd and k  5. • We have a cyclic 3-sun system of order n 13 (mod 24). 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  24. 0 1  2 3 10 4 9 8 5 6 n 0 (mod 12) • n = 12 • V(K12) = {}Z11 • difference set (K11)={1,2,...,5} • Base S(C3) : [(1, 2, 4), 5, 7, ] [(2, 3, 5), 6, 8, ] [(3, 4, 6), 7, 9, ] [(4, 5, 7), 8, 10, ] … 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會 7

  25. Theorem: (1) If n 1 (mod 12),Kncan be decomposed into cyclic3-sun system. (2) If n 0 (mod 12),Kncan be decomposed into 1-rotational 3-sun system. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會 Danshui

  26. n 9 (mod 12) • There exists KTS(v)  v  3 (mod 6) • n 9 (mod 12)  there exists KTS(n) • Example: n = 9 • Construct KTS(9)1, 2, 3 1, 4, 7 1, 5, 9 1, 6, 8 4, 5, 6 2, 5, 8 2, 6, 7 2, 4, 97, 8, 9 3, 6, 9 3, 4, 8 3, 5, 7 • (1, 2, 3); 4, 5, 6 (1, 5, 9); 6, 7, 2 (4, 5, 6); 7, 8, 9 (2, 6, 7); 4, 8, 3 (7, 8, 9); 1, 2, 3 (3, 4, 8); 5, 9, 1 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  27. 1 3 2 4 6 5 7 9 8 • 1, 2, 3 1, 4, 7 (1, 2, 3); 4, 5, 6 • 4, 5, 6 2, 5, 8 (4, 5, 6); 7, 8, 9 7, 8, 9 3, 6, 9 (7, 8, 9); 1, 2, 3 1 7 4 2 8 5 3 9 6 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  28. Result • If n  9 (mod 12), then there is a 3-sun system of order n. 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  29. Embed a cyclic STS into a 3-sun system 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  30. STS(7) is embedded into 3-sun system of order 13 8, 9, 10 10, 9, 8 10, 9, 11 12, 9, 8 10, 13, 9 10, 9, 11 12, 10, 11 1, 2, 4 2, 3, 5 3, 4, 6 4, 5, 7 5, 6, 1 6, 7, 2 7, 1, 3 6, 13, 4 6, 7, 5 8, 5, 4 7, 8, 13 6, 5, 8 7, 4, 12 8, 9, 11 9, 10, 12 10, 11, 13 11, 12, 1 12, 13, 2 13, 8, 3 K7 K13 13*12/12=13, 7*6/6=7 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  31. Result • Let m be a positive integer. • Let (X, T) be a cyclic Steiner triple system of order 6m+1. • Then there is a 3-sun system (Y, S) of order 12m+1, such that (X, T) is embedded in (Y, S). 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

  32. The End 謝 謝 ! Thank you for your attention! 2011年圖論與組合學研討會暨第六屆海峽兩岸圖論與組合學研討會

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