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R ainbow connection numbers of Cartesian product of graphs

R ainbow connection numbers of Cartesian product of graphs. Yu-Jung Liang Department of Applied Mathematics National Dong Hwa University Advisor: Professor David Kuo. Outline. Introduction Previous Results Main Results. Definition (The Cartesian product).

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R ainbow connection numbers of Cartesian product of graphs

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  1. Rainbow connection numbers of Cartesian product of graphs Yu-Jung Liang Department of Applied Mathematics National Dong Hwa University Advisor: Professor David Kuo

  2. Outline • Introduction • Previous Results • Main Results

  3. Definition (The Cartesian product) • Give two graphs and , the Cartesian product of and , denoted by , is defined as follows: .Two distinct vertices and of are adjacent if and only if either and or and .

  4. Example:

  5. Definition (Rainbow connection number) • A path is rainbow if no two edges of it are colored the same. • An edge-coloring graph is rainbow connected if any two vertices are connected by a rainbow path. • We define the rainbow connection number of a connected graph , denoted by rc, as the smallest number of colors that are needed in order to make rainbow connected. • A graph is strong rainbow connected if there exists a rainbow geodesic for any two vertices and in .

  6. 2 5 • Example: 2 1 3 1 2 1 3 4 3 1 2 1 2 3 1 2 4 5 1 2 3 1 1 2 3 4 2 5 rc

  7. Previous Results • Theorem 1 (Xueliang Li, Yuefang Sun) For any connected graph . • Theorem 2 (XueliangLi, YuefangSun) Let , where each is connected. Then we have . Moreover, if for each , then the equality holds. back1

  8. Theorem 3 (M. Basavaraju, L.S. Chandran, D. Rajendraprasad, and A. Ramaswamy) If and are non-trivial connected graphs, then . back

  9. Main Results • Particular labeling 0 1 2 3 1 3 4 0 1 2 0 1 2 3 0 1 2 3 3 4 0 1 0 2 0 1 2 3

  10. Rainbow connection numbers of the Cartesian product of paths and cycles • If , then for all . • If , then for all . • If n is even, then , for all .

  11. 3 • Example: 2 2 1 3 1 1 3 3 1 2 1 2 1 2 1 2 3 1 1 2 3 Thm1

  12. Rainbow connection numbers of the Cartesian product of two trees If and are trees, we have follows conclusion about .

  13. If , , then . 1 1 3 2 3 1 3 2 Thm3

  14. If , has three edge-disjoint path with length , then .

  15. 7 • Otherwise, we have . Example 1 1 4 4 5 2 2 3 3 6 6

  16. 7 4 4 1 1 2 5 5 3 3 6 6

  17. 7 4 4 1 1 • Example 7 7 5 2 5 8 8 8 7 8 6 6 3 3 1 1 4 4 8 8 8 5 2 2 8 8 8 3 3 6

  18. 7 8 1 1 • Example 1 4 4 2 5 5 2 2 3 3 3 3 6 6 6

  19. 7 8 4 4 4 1 1 5 2 2 5 5 6 6 6 6 3 3 3

  20. 8 7 4 4 8 • Example 5 4 1 1 7 8 7 1 9 9 7 5 5 9 6 6 1 4 9 9 2 9 2 2 6 6 9 5 9 9 3 2 3 3 9 9 3 9 9 9 9 9 3 6 6

  21. 1 • Example 2 1 3 3 2 1 2 3 1

  22. 2 1 • In fact, if we consider subgraph of , we have three nature graph. Hence easy to check have rainbow path for any vertices in . 3 2 1 2 1 1 2 2 3 1 3 3 2 1 3 2 1 2 3 2 3 1 3 1 1 2 1 3 1

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