1 / 32

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.

amato
Download Presentation

From Steiner Triple Systems to 3-sun systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related