the spanning laceability of k ary the spanning
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The spanning laceability of k-ary The spanning laceability of k ary - PowerPoint PPT Presentation

The spanning laceability of k-ary The spanning laceability of k ary n-cubes when k is even : : 1 Outline Introd ction Introduction Preliminaries Definition Theorem 1


  1. The spanning laceability of k-ary The spanning laceability of k ary n-cubes when k is even 學生:張書莉 指導教授 高欣欣老師 指導教授:高欣欣老師 1

  2. Outline Introd ction Introduction Preliminaries Definition Theorem 1 Theorem 2 M i Main results lt Lemma 1-5 Theorem 3 2

  3. Paper p1 Introduction (0,0) (1,0) (2,0) (3,0) (4,0) (5,0) 0 (0,1) (1,1) (2,1) (3,1) (4,1) (5,1) 1 (0,2) (1,2) (2,2) (3,2) (4,2) (5,2) 2 (0,3) (1,3) (2,3) (3,3) (4,3) (5,3) 3 (0,0,0) (0,1,0) (1,0,0) (1,1,0) 0 (0,0) (1,0) (0,4) (1,4) (2,4) (3,4) (4,4) (5,4) 4 (0,1) (0,1) 1 1 (1 1) (1,1) (0,0,1) (0,1,1) (1,0,1) (1,1,1) (0,5) (1,5) (2,5) (3,5) (4,5) (5,5) 2 5 2 2 Q Q Q 1 2 3 6 6 Q Q 1 2 hypercube k-ary n-cube 3

  4. Introduction 6 k ( ( ) ) Q Q Q Q 3 n 6-neighbers (2n-neighbers) − u 1 1 k k 1 1 u u 4-neighbers (2n-2) − 6 , 0 k , 0 6 , 1 k , 1 6 , 5 k , k 1 Q Q ( ( Q Q ) ) Q Q ( ( Q Q ) ) Q Q ( ( Q Q ) ) − − − 2 2 n n 1 1 2 2 n n 1 1 2 2 n n 1 1 4

  5. Paper p1 Introduction Recent researches 1. 1. , b b w w 4 4 Q Q edge-bipancyclic edge-bipanconnected 2 2 2. 3. P1 P1 v v v u u P1 P2 P2 5 5 Q Q 2*-connected 2-connected 2 2 5

  6. 6 b Introduction k n Q w Paper p2

  7. Paper p3 Preliminaries 1. u 1*-connected v 1*-laceable 2. u 2*-connected v v 2*-laceable 7

  8. 8 Preliminaries B B W W Paper p3 3. 4.

  9. Preliminaries Paper p3 C(u,v) is k-container, k*-container; G is k-connected, k*-connected 5. u u u3 u1 u2 C(u,v) is 3*-cantainer v 9

  10. Paper p4 Preliminaries 6. (0,0) (1,0) (2,0) (3,0) (4,0) (5,0) (0,1) (1,1) (2,1) (3,1) (4,1) (5,1) (0,2) (1,2) (2,2) (3,2) (4,2) (5,2) 6 Q 2 1th-bit ( , ) (0,3) ( , ) (1,3) (2,3) ( , ) ( , ) (3,3) ( , ) (4,3) ( , ) (5,3) (0,4) (1,4) (2,4) (3,4) (4,4) (5,4) 1th bit 1th-bit (0,5) (1,5) (2,5) (3,5) (4,5) (5,5) 0th-bit 0th-bit 10

  11. 11 Preliminaries Preliminaries Paper p4

  12. Main results Main results Lemma 1 tools tools Lemma 2 Lemma 3 Lemma 3 k k Q Base cases for n Lemma 4 mathematical induction mathematical induction k Q 2 proof Lemma 5 k k Q Q Theorem3 Theorem3 p proof n n k 4 6 8 n Lemma 5 Lemma 4 2 Lemma 3 3 n n-1 1 n 2 2 Theorem 3 n-1 12 n

  13. Main results Paper p5 w b − + Q , k j k , j ' 1 k , j ' k , j 1 Q Q − Q − − − n 1 n 1 n 1 n 1 Proof: 13

  14. 14 b 1 1 Q − , k n 1 x Lemma 1 Lemma 1 0 1 b Q − , 1 j j k n − Q , k k n 0 Q w x w Paper p5 Proof:

  15. Paper p5 Lemma 1 Lemma 1 Proof: ' − j 1 w + y y j 1 j ' y y y y ' − j ' 1 1 + x j j 1 x x b + − k , j 1 k , j ' Q , k j k , j ' 1 Q Q Q − Q Q Q Q − − n n 1 1 − n n 1 1 n 1 1 n 1 1 15

  16. Paper p6 Main results Main results w b + − Q , k i k , i 1 k , j 1 Q , k j Q Q − − − − n 1 n 1 n 1 n 1 Proof: The same as Lemma 1-case1 Similar to case2.1 16

  17. Lemma 2 Paper p6 Proof: w + + − 1 i i 1 i j 1 j − z x x z x j 1 x + + − i i 1 i 1 j j 1 − y y w y y j 1 w b b 17 + − k , i 1 Q , k j Q , k i k , j 1 Q Q − − − − n 1 n 1 n 1 n 1

  18. Paper p7 Lemma 3 4 Lemma 3, 4 w & b respective direction b1 b3 w b1 b2 b2 b4 b4 4 6 Q Q 2 2 18

  19. 19 b2 b2 w w w 4*-laceable 3* laceable 3*-laceable b1 b1 w w Proof: Paper p7

  20. 20 b4 b4 w w b3 b3 w w 3*-laceable 4*-laceable b2 b2 w w Paper p8-10 Proof: b1 b1 w w

  21. 21 6 Q 2 2 Lemma 5 Q reuglar Paper p11-17

  22. 22 Lemma 5 Lemma 5 Paper p11-17

  23. Lemma 5 Paper p11-17 case1 case2 case3 case4 = − ≤ − ≤ − = k − v k 1 for some v t k 2 v t k 2 Case 4 v 1 Case 1 Case 2 Case 3 − − − t 1 1 1 0 ≥ = t 2 and v 0 ∈ ∈ ∉ ∉ E E ( ( C C ( ( w w , b b )) )) E E ( ( C C ( ( w w , b b )) )) o ((k 2 k 1) (k 1 k 1)) ((k-2,k-1),(k-1,k-1)) ((k 2 k 1) (k 1 k 1)) ((k-2,k-1),(k-1,k-1)) = = = = case 1.1 s 1 case 2.1 t 1 case 4.1 t 2 case 3.1 t 1 ≥ = = = case 1.2 s 2 case 2.1.1 s 1 case 4.1.1 s 1 case 3.1.1 s 1 = ≥ ≥ case 2.1.2 s 2 case 4.1.2 s 2 case 3.1.2 s 2 ≥ ≥ ≥ case 2.1.3 s 3 case 4.2 t 3 case 3.2 t 2 ≥ ≥ = = case 2 2 2.2 t t 2 2 case 4 2 1 4.2.1 s 1 1 case 3.2.1 s 1 = ≥ ≥ case 2.2.1 s 1 case 4.2.2 s 2 case 3.2.2 s 2 = case 2.2.2 s 2 ≥ ≥ case case 2 2 3 2.2.3 s s 3 3 23

  24. 24 step3 8 2 2 Q b w Lemma 5 step2 p f regular step1 b 6 2 2 Q Q Paper p11-17 w Proof:

  25. 25 Theorem 3 Paper p17 Proof:

  26. Paper p17 P + m 1 − 1 k 1 P w w w m − m 1 { P } = i i 0 1 b − k 1 b b − − k k , 1 1 k k , k k 1 1 k k , 0 0 k k , 2 2 k k , k k 2 2 Q − Q Q − Q − Q − − n 1 n 1 n 1 n 1 n 1 26

  27. Paper p17-20 m = P P 1 = − 2 k 1 P P b w + b w m 1 2 P 1 z z 0 − k , 1 k , k 1 k , 0 k , 2 Q − Q Q − Q − − n 1 n 1 n 1 n 1 27

  28. Paper p17-20 P m − 2 k 1 P b w + w b m 1 − − 1 m 1 } m 1 { S } { S = = i i 0 0 i i 1 1 1 1 y y y y − − m 1 0 m 1 y 0 1 z z P 0 , − k , 0 k , 1 k k 1 k , 2 Q Q − Q − Q Q Q Q Q − − n 1 n 1 n 1 n 1 28

  29. Paper p17-20 P 2 − 2 k 1 w b b w P 1 1 w 1 P z z 0 − − k , 0 k , 1 k , 2 k , k 2 k , k 1 Q − Q − Q − Q Q − − n 1 n 1 n 1 n 1 n 1 29

  30. Paper p17-20 P 2 − k 1 w 2 b w b − 0 k 2 P x g 3 0 e P 1 − k 1 e P 1 w 0 0 b g x f − − ' } m 2 0 ' } m 2 { { T T } { { S S } = = i i 1 i i 1 0 0 0 1 y y y y y y y y 1 1 − y − 0 1 m 1 m 1 0 − k 1 f 1 z z , − , − k , 1 k k 1 k , 0 k , 2 k k 2 Q Q − Q Q Q Q − Q Q − Q Q − − n 1 n 1 n 1 n 1 n 1 30

  31. Paper p17-20 P 2 P − b k 2 − w 2 g k 1 1 b w 0 1 x x 0 0 e P 3 P P 0 − b 1 k 1 0 e w 0 x g f − − 0 ' } 1 ' } m 1 m { T { S = = i i 1 i i 1 0 0 y y 0 0 − k 1 1 f f z z z z − − k , 0 k , 1 k , 2 k , k 2 k , k 1 Q − Q − Q − Q Q − − n 1 n 1 n 1 n 1 n 1 31

  32. Paper p21 P + m 1 ' + − 1 j ' j 1 k 1 P w w w w w m 0 1 1 j ' x x x x − m 1 0 0 0 0 0 0 − − 1 } 1 } m m 1 1 − { { U U 0 } 0 } m m 1 1 j j ' ' } m 1 1 { { U = { U = p p 0 = p p 0 p p 0 0 y 0 j ' j ' y 1 1 y y y y 0 − m 1 − − 0 0 0 1 m 1 m 1 b b b ' + − j 1 k 1 b b + , − , − k , 0 k , j ' k , j ' 1 k k 2 k k 1 k , 1 Q − Q Q Q − Q Q Q Q Q Q Q − Q − − − n 1 n 1 n 1 n 1 n 1 n 1 32

  33. Paper p22-24 P 2 − − ' + − − 1 j 1 j 1 k 3 k 2 k 1 b P w b b b b w w 0 − k 2 ' + f j 1 f − k k 3 3 f f j ' f e − k k 1 1 b b − k 1 − − z k 3 k 2 e e z ' + j 1 e P 1 − − − k , j ' 1 k , k 2 k , k 1 + Q Q Q k , 0 k , 1 k , j ' k , j ' 1 Q − Q − Q − Q − − − n 1 n 1 n 1 − 1 1 n n 1 n n 1 33

  34. Paper p22-24 P P 0 1 P w 2 ' + − j 1 w k 1 b P w w b 1 2 P 2 x 3 0 1 x ( ( 1 ) ) x x ( ( 1 1 ) ) 0 0 0 0 ( 2 ) x 1 x ( 2 ) 0 0 1 x ( 3 ) x ( 3 ) 0 0 1 x ( 4 ) x ( 4 ) 0 0 − m 2 − − − { Y } m 1 1 } m 1 2 } m 1 { S } { S { U = 0 p p = = = p p 1 p p 1 p p 0 0 α α − 0 α α − 1 x x ( ( 1 1 ) ) x x ( ( 1 1 ) ) 0 j ' t 2 t − 0 α 1 j ' m 2 0 α x ( ) t 0 x ( ) 0 2 b z = 0 1 2 j ' z z z z = k , 4 − k k , , j j ' 3 3 Q Q − k k , 0 0 k k , 1 1 k k , 2 2 k k , k k 1 1 Q Q Q Q − Q − Q Q − Q Q Q − n 1 − n 1 n 1 n 1 n 1 n 1 34

  35. conclusion Lemma 3 mathematical induction k Q Base cases for n n k Lemma 4 k Q 2 Q 2 proof p Lemma 5 Lemma 5 k Q Theorem3 proof n n 2 n-1 n k Q , there exist 2n Given any pair of vertices, w and b, from different partite sets of n − 2 n 1 k Q Internally disjoint paths between w and b, such that U P covers all vertices of n i = i 0 ≥ ≥ ≥ ≥ 2 2 n n k k 4 4 for any even integer for any even integer and any integer and any integer . . 35

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