Hamiltonian Paths and Cycles in Planar Graphs

1. Hamiltonian Paths and Cycles in Planar Graphs Sudip Biswas1, Stephane Durocher2, Debajyoti Mondal2 and Rahnuma Islam Nishat3 • 1Department of Computer Science, Louisiana State University, USA • 2Department of Computer Science, University of Manitoba, Canada • 3Department of Computer Science, University of Victoria, Canada

2. Problem Definition Planar Graph COCOA ’12, Banff August 06, 2012

3. Problem Definition O(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. Planar Graph COCOA ’12, Banff August 06, 2012

4. Problem Definition O(n1.46557n) upper bound and Ω(1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph Outerplanar Graph COCOA ’12, Banff August 06, 2012

5. Previous Works Eppstein (2004) Gebauer (2011) O(2nn4) time and O(n3log n) space algorithm to count the Hamiltonian paths and cycles in a given graph. Algorithm to count the number of Hamiltonian cycles in m×n grid graphs for m=1,2,3,4,5. Count all Hamiltonian cycles in a degree three graph in time O(23n/8) ≈ 1.297n O(1.783n) upper bound on the number of Hamiltonian cycles in 4-regular graphs Eric T. Bax (1993) Collins and Krompart (1997) COCOA ’12, Banff August 06, 2012

6. Previous Works • O(2.3404n) upper bound and Ω(2.0845n) lower bound on the number of Hamiltonian cycles in planar graphs. • O(2.8927n) upper bound and a Ω(2.4262n) lower bound on the number of simple cycles in planar graphs. Buchin et al. (2007) Θ(1.502837 n) bound on the number of simple cycles in outerplanar graphs. de Mier and Noy(2009) COCOA ’12, Banff August 06, 2012

7. Our Results O(n1.46557n) upper bound and Ω(1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph For any positive integer n ≥ 6, we define an outerplanar graph G, called a ZigZagouterplanar graph, such that the numberof Hamiltonian paths starting at a single vertex in G is the maximum. O(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA ’12, Banff August 06, 2012

8. Counting Hamiltonian Paths a a c b c b k2 k2 k1 k1 d d • T(n) = T(n-k2-2) COCOA ’12, Banff August 06, 2012

9. Counting Hamiltonian Paths a a c b c b k2 k2 k1 k1 d d • T(n) = +T(n-k2-3) T(n-k2-2) COCOA ’12, Banff August 06, 2012

10. Counting Hamiltonian Paths a a c b c b k2 k2 k1 k1 d d • T(n) = +T(n-k2-3) +T(n-k1-2) T(n-k2-2) COCOA ’12, Banff August 06, 2012

11. Counting Hamiltonian Paths a a c b c b k2 k2 k1 k1 d d • T(n) = +T(n-k2-3) +T(n-k1-2) +T(n-k1-3) T(n-k2-2) COCOA ’12, Banff August 06, 2012

12. Counting Hamiltonian Paths a a c b c b k2 k1 k1 d d • T(n) = T(n-1) + T(n-3) COCOA ’12, Banff August 06, 2012

13. Counting Hamiltonian Paths a a c c b b k1 k1 k2 k2 • T(n) = T(k2+2) + T(k1+2) COCOA ’12, Banff August 06, 2012

14. Counting Hamiltonian Paths • T(n) = max{T(n-k2-2) + T(n-k2-3) + T(n-k1-2) + T(n-k1-3), • T(n-1) + T(n-3), • T(k2+2) + T(k1+2) } • = T(n-1) + T(n-3) • = O(1.46557n) COCOA ’12, Banff August 06, 2012

15. Our Results O(n1.46557n) upper bound and Ω(1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph We call this vertex an ace vertex of G For any positive integer n ≥ 6, we define an outerplanar graph G, called a ZigZagouterplanar graph, such that the number of Hamiltonian paths starting at a single vertex in G is the maximum. O(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA ’12, Banff August 06, 2012

16. Maximum Hamiltonian Paths Starting from a Vertex Outerplanar Graphs with 7 vertices Step 1: Let G be the outerplanar graph such that the number of Hamiltonian paths starting from a vertex of G is the maximum among all outerplanar graphs with the same number of vertices. Then the weak dual of G is a path. COCOA ’12, Banff August 06, 2012

17. Child Swap Operation Let x be an ace vertex and let x be in G1. G1 b a v Child swap operation does not decrease the number of Hamiltonian paths starting from x. G2 G3 c b’ a’ u c’ COCOA ’12, Banff August 06, 2012

18. Repeated Ancestry v is the left child of u w is the left child of v x a b u e v w d c y COCOA ’12, Banff August 06, 2012

19. Child Flip and Parent Flip v is the left child of u w is the left child of v If ace vertex is not e apply child flip Otherwise apply parent flip a e v x x w a a b b d c u u y a’ b’ v e e v z u w w b c’ x c d d c y y COCOA ’12, Banff August 06, 2012

20. ZigZag Graph a b Ace vertex • T(n) = T(n-1) + T(n-3) O(n1.46557n) upper bound and Ω(1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph z y COCOA ’12, Banff August 06, 2012

21. Our Results O(n1.46557n) upper bound and Ω(1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph For any positive integer n ≥ 6, we define an outerplanar graph G, called a ZigZagouterplanar graph, such that the number of Hamiltonian paths starting at a single vertex in G is the maximum. O(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA ’12, Banff August 06, 2012

22. Cycle-Paths in Planar Graphs A cycle-path is a simple path that can be extended to a simple cycle. b Split the edges incident to each vertex into two sets i d e j f c a k g COCOA ’12, Banff August 06, 2012

23. Cycle-Paths in Planar Graphs A cycle-path is a simple path that can be extended to a simple cycle. b {a} i {a,b} {a,d} d e j f c a k g COCOA ’12, Banff August 06, 2012

24. Cycle-Paths in Planar Graphs A cycle-path is a simple path that can be extended to a simple cycle. b {a} i {a,b} {a,d} d {a,b,j} {a,b,i} e {a,b,k} j f c a k g COCOA ’12, Banff August 06, 2012

25. Cycle-Paths in Planar Graphs A cycle-path is a simple path that can be extended to a simple cycle. b {a} i {a,b} {a,d} . . . d {a,b,j} {a,b,i} e {a,b,k} j f . . . . . . c a k g P(n,f) ≤ kv.P(n-1,f-kv+1) +1 P(n,f) = O(2n) COCOA ’12, Banff August 06, 2012

26. Extending Perfect Matching to a Cycle We start with a perfect matching and complete it into a cycle. b {a,b} i {a,b,i,e} {a,b,j,k} {a,b,k,j} . . . . . . . . . d e Starting from one edge, P(n,f) = O(2n/2) For O(n) edges, the number of Hamiltonian cycles is ≤ O(n2n/2) j f c a k g COCOA ’12, Banff August 06, 2012

27. Upper Bound on the Number of Hamiltonian Cycles Each perfect matching can be extended to O(n).2n/2 Hamiltonian cycles. Number of perfect matchings is bounded by 6n/4. Therefore, number of Hamiltonian cycles in planar graph is 6n/4 × O(n). 2n/2 < O(n).2.2134n COCOA ’12, Banff August 06, 2012

28. Summary of Our Results O(n1.46557n) upper bound and Ω(1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph For any positive integer n ≥ 6, we define an outerplanar graph G, called a ZigZagouterplanar graph, such that the number of Hamiltonian paths starting at a single vertex in G is the maximum. O(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA ’12, Banff August 06, 2012

29. Open Problems • Can we extend the techniques used in this paper to bound the number of Hamiltonian paths and cycles in planar graphs with bounded treewidth? • We calculated the upper bound on planar graphs under certain assumptions on the recursion tree. Is there an alternative proof without these assumptions? COCOA ’12, Banff August 06, 2012

30. Thank You