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

Introduction Introduction □ G is k-or de r e d □ for any sequence of k distinct vertices v 1 ,v 2 …v k of G there exists a cycle in G containing these k vertices in the specified order. G v 1 v 2 v 3 v 4 4 4-ordered d d

Introduction (cont ) Introduction (cont.) □ G is k-or de r e d Hamiltonian □ If G is k-ordered and the required cycle is Hamiltonian. G v 1 v 2 v 3 v 4 4 ordered Hamiltonian 4-ordered Hamiltonian

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 v 1 ,v 2 …v k of 1 2 k G there exists a Hamiltonian path in G containing these k vertices in the specified order. This path start from v 1 and end to the v k . t t f d d t th G v 1 v 2 v 2 v 3 v 4 4-ordered Hamiltonian connected

Introduction (cont ) Introduction (cont.) □ G is k-or de r e d Hamiltonian lac e able □ G is a bipartite graphs. G v v 1 v 2 v 3 v 3 v 4 4-ordered Hamiltonian laceable

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. □ Problem 3. Determine the best possible degree P bl 3 D t i th b t ibl d 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.

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 Heawood graph 3-regular 4-ordered graphs 3 l 4 d d h 3 3-regular 4-ordered Hamiltonian graphs l 4 d d H ilt i h

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)

Our Results

Generalized honeycomb torus Generalized honeycomb torus GHT(3,8,1) ( ) GHT(4,8,0) ( ) GHT(4,8,2) ( )

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

Generalized Petersen graphs Generalized Petersen graphs P(8,1) P(8,2) P(8,3)

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}.

Chordal ring Chordal ring 13 0 13 0 12 1 12 1 11 2 11 2 10 10 3 3 9 9 9 9 4 4 8 5 8 5 7 6 7 6 CR(14,1,5) CR(14,1,3) =CR 14 (1,-1,5) =CR 14 (1,-1,5)

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.

cubic 4-ordered Cells cubic 4 ordered Cells p 1 f (p 1 ) (p 1 ) q 1 f (q 1 ) G 1 G 1 G 2 G 2 f (r 1 ) r 1 f (s 1 ) ( 1 ) s 1 1 O f (C 1 , C 2 ) f ( 2 ) 1

cubic 4 ordered graphs cubic 4-ordered graphs □ For example ： GHT(3,6,1) GHT(3,8,1)

cubic 4 ordered graphs cubic 4-ordered graphs Heawood graph GHT(3,8,1) ( , , )

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)

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)

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.

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 odd odd.

T ha nks for your liste ning !! Q & A