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Outline Outline
1 I t d ti
- 1. Introduction
- 2. Our Results
- 3. Q&A
On cubic 4-ordered graphs and cubic 4-ordered Hamiltonian graphs - - PowerPoint PPT Presentation
On cubic 4-ordered graphs and cubic 4-ordered Hamiltonian graphs - - PowerPoint PPT Presentation
On cubic 4-ordered graphs and cubic 4-ordered Hamiltonian graphs Hamiltonian graphs Lih-Hsing Hsu Speaker Ming Tsai Speaker Ming Tsai Outline Outline 1 I t 1. Introduction d ti 2. Our Results 3. Q&A Introduction
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SLIDE 3
Introduction
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Introduction Introduction
□ G is k-or
de r e d
□ for any sequence of k distinct vertices v1,v2…vk of G there exists a cycle in G containing these k vertices in the specified order.
v1 v2 G v3 v4 4 d d 4-ordered
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Introduction (cont ) Introduction (cont.)
□ G is k-or
de r e d Hamiltonian
□ If G is k-ordered and the required cycle is Hamiltonian.
v1 v2 G v3 v4 4 ordered Hamiltonian 4-ordered Hamiltonian
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Introduction (cont ) Introduction (cont.)
□ G is k-or
de r e d Hamiltonian c onne c te d
□ for any sequence of k distinct vertices v1,v2…vk of
1 2 k
G there exists a Hamiltonian path in G containing these k vertices in the specified order. This path t t f d d t th start from v1 and end to the vk.
v1 v2 G v2 v3 v4 4-ordered Hamiltonian connected
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Introduction (cont ) Introduction (cont.)
□ G is k-or
de r e d Hamiltonian lac e able
□ G is a bipartite graphs.
v G v1 v2 v3 v3 v4 4-ordered Hamiltonian laceable
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Introduction (cont ) Introduction (cont.)
□ L. Ng, M. Schultz, k-Orde re d hamilto nian
- graphs. J. Graph Theory 24 (1997) 45-57
□ Problem 1. Determine the best possible degree condition for Theorem 4. □ Problem 2. Determine whether there is an infinite class of 3-regular 4-ordered graphs. P bl 3 D t i th b t ibl d □ Problem 3. Determine the best possible degree condition for Theorem 14. Problem 4 Study the existence of small degree k □ Problem 4. Study the existence of small degree k- Hamiltonian-connected graphs.
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Introduction (cont ) Introduction (cont.)
□ K. Meszaros, On 3-re gular 4-o rde re d graphs. Disc re te Math. 308 (2008) 2149-2155.
Petersen graph 3 l 4 d d h Heawood graph 3 l 4 d d H ilt i h 3-regular 4-ordered graphs 3-regular 4-ordered Hamiltonian graphs
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Introduction (cont ) Introduction (cont.)
□ Generalized Honeycomb torus GHT(3,n,1) is 4-ordered for any even integer n with n ≥ 8.
GHT(3 8 1) GHT(3,8,1)
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Our Results
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Generalized honeycomb torus Generalized honeycomb torus
GHT(3,8,1) GHT(4,8,0) GHT(4,8,2) ( ) ( ) ( )
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cubic 4 ordered graphs cubic 4-ordered graphs
□ Assume that m is an odd integer with m≧3 and n is an even integer with n≧4. The generalized honeycomb tours GHT(m,n,1) is 4-ordered if and only if n≠4 □ Assume that m is an even with m≧2 and n is □ Assume that m is an even with m≧2 and n is an even integer with n≧4. The generalized honeycomb tours GHT(m,n,0) is 4-ordered if honeycomb tours GHT(m,n,0) is 4 ordered if and only if m≠ 2 and n≠ 4
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Generalized Petersen graphs Generalized Petersen graphs
P(8,1) P(8,2) P(8,3)
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cubic 4 ordered graphs cubic 4-ordered graphs
□ P(n,1) is not 4-ordered □ P(n,2) is not 4-ordered if n ≠ 5. □ P(n,3) is 4-ordered if n ≥ 7 unless n {7, 9, 12}.
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Chordal ring Chordal ring
1 12 13 1 12 13 2 11 2 11 3 9 10 3 9 10 4 5 6 7 8 9 4 5 6 7 8 9 CR(14,1,5) =CR14(1,-1,5) CR(14,1,3) =CR14(1,-1,5)
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cubic 4 ordered graphs cubic 4-ordered graphs
□ Computer program result：
□ CR(n,1,k) is 4-ordered if 5 ≦ k ＜ n/2 -1 and n is even.
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cubic 4 ordered Cells cubic 4-ordered Cells
p1 f (p1) q1 (p1) f (q1)
G1 G2
r1
s1 f (r1) f (s1)
G1 G2
1
( 1)
Of (C1, C2)
f ( 1 2)
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cubic 4 ordered graphs cubic 4-ordered graphs
□ For example：
GHT(3,6,1) GHT(3,8,1)
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cubic 4 ordered graphs cubic 4-ordered graphs
GHT(3,8,1) Heawood graph ( , , )
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cubic 4-ordered Hamiltonian graphs
□ P(n,3) is 4-ordered Hamiltonian if and only if n is even and either n = 18 or n ≥ 24.
P(24 3) P(24,3)
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cubic 4-ordered Hamiltonian graphs
□ CR(n,1,5) is 4-ordered Hamiltonian graph if n=12k+2 and n=12k+10 with k ≧2 and n = 14.
CR(26 1 5) CR(26,1,5)
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cubic 4-ordered Hamiltonian laceable graphs
□ Computer program result：
□ P(n,3) is 4-ordered Hamiltonian laceable when n is even and 10 ≦ n ≦ 52. □ CR(2n, 1, 5) is 4-ordered Hamiltonian laceable when 38 ≦ n ≦ 92 and n≠4t+2.
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cubic 4-ordered Hamiltonian connected graphs
□ Computer program result：
□ P(n,3) is 4-ordered Hamiltonian laceable when n is even and 10 ≦ n ≦ 52. □ P(n,3) is 4-ordered Hamiltonian connected when n is odd and 19 ≦ n ≦ 47. ※ P(n,3) is not 4-ordered Hamiltonian when n is
- dd
- dd.
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