1. All-to-all broadcast problems on Cartesian product graphs Jen-Chun Lin 林仁俊 指導教授：郭大衛教授 國立東華大學 應用數學系碩士班

2. Outline Introduction Main result Reference

3. Introduction

4. Suppose each vertex has a private message needed to send to every other vertices(all-to-all broadcast). At each time unit, vertices exchange their messages under the following constraints: (1) only one message can travel a link at a time unit (2) a message requires one time unit to be transferred between two nodes (3) each vertex can use all of its links at the same time

5. Chang et al. They gave upper and lower bounds for all-to- all broadcast number (the shortest time needed to complete the all-to-all broadcast) of graphs and give formulas for the all-to-all broadcast number of trees, complete bipartite graphs and double loop networks under this model.

6. All-to-all broadcasting numbers of Cartesian product of cycles and complete graphs

7. Given a graph G and a positive integer t, we use G ◦ t to denote the multigraph obtained from G by replacing each edge in E(G) with k edges

8. ① ② ③ ① ② ③ ① ①② ② ③ ③

10. For a graph G with a set of edge-disjoint subgraphs of G◦t is a t-broadcasting system of G.

11. Theorem

13. 0,0 2,0 1,0 0,1 1,1 1,2 0,3 3,0 0,4 1,3 1,42,4 2,2 2,3 3,1 3,4 3,3 3,2 2,1 0,2 ① ① ② ② ③ ③ ④ ④ ⑤ ⑤ ⑥ ⑥⑦ ⑦ ⑧ ⑧ ⑨ ⑨ ⑩ The broadcasting tree

14. 0,0 2,0 1,0 0,1 1,1 1,2 0,3 3,0 0,4 1,3 1,42,4 2,2 2,3 3,1 3,4 3,3 3,2 2,1 0,2 ① ① ② ② ⑨ ⑧ ④ ⑤⑥ ③ ⑨ ③ ④ ⑧ ⑩ ⑦ ⑤⑥ ⑦ The broadcasting tree

15. Lemma Given a graph G, if and only if there exists a t-broadcasting systm of G.

16. Lemma For any graph G with,

17. Theorem

20. 0,0 2,0 1,0 0,1 1,1 1,2 0,3 3,0 0,4 1,3 1,42,4 2,2 2,3 3,1 3,4 3,3 3,2 2,1 0,2 4,0 4,1 4,4 4,3 4,2 The perfect broadcasting tree ① ② ③④ ⑤⑥ ⑦⑧ ① ① ② ② ③ ③ ④ ④ ⑤ ⑤⑥ ⑥ ⑦ ⑦ ⑧ ⑧

21. Theorem For positive integers m and n with We set

24. 3 1 The subgraph 0,0 0,2 0,1 0,4 0,3 6,0 3,0 2,0 1,0 7,0 4,0 8,0 5,0 1 1 2 2 111 222 2 333 3 3 4 444 4 4 5 555 5 5 6 666 6 6

25. 0,0 0,1 6,0 3,0 2,0 1,0 7,0 4,0 8,0 5,0 0,2 0,4 0,3 7 7 7 7 7 7 8 8

26. All-to-all broadcasting numbers of hypercubes

27. Theorem

28. For a vertex the weight of, denoted by, is defined by Definition 0 3 2 1 n個n個

29. For the set the right-shift of elements in is a function from to defined By = Definition =ex:

30. the orbit of v, denoted by A vertex v in is said to be Definition

32. Theorem

33. Lemma (0,0,………,0) (1,0,………,0)(0,1,………,0) (0,0,………,1) ……… 方向 1 方向 2 方向 n (2,0,………,0)(0,2,………,0) (0,0,………,2) (1,0,………,1)(1,1,………,0)

34. Lemma (1,1,………,1) (0,1,………,1)

35. 6 2 0,0 0,11,0 0,2 1,1 2,0 0,31,23,0 2,1 3,1 1,3 2,2 3,2 2,3 3,3 11 2 4 433 55 6 7 7 8

36. Thanks for your listening!