Sudip Biswas 1, Stephane Durocher 2, Debajyoti Mondal 2 and Rahnuma Islam Nishat 3 Hamiltonian Paths and Cycles in Planar Graphs 1 Department of Computer Science, Louisiana State University, USA 2 Department of Computer Science, University of Manitoba, Canada 3 Department of Computer Science, University of Victoria, Canada

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

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

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

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

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

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

Counting Hamiltonian Paths a b c d k1k1 k2k2 a b c d k1k1 k2k2 T(n) = T(n-k 2 -2) COCOA ’12, Banff August 06,

a b c d k1k1 k2k2 Counting Hamiltonian Paths a b c d k1k1 k2k2 T(n) = T(n-k 2 -2) +T(n-k 2 -3) COCOA ’12, Banff August 06,

a b c d k1k1 k2k2 Counting Hamiltonian Paths a b c d k1k1 k2k2 T(n) = T(n-k 2 -2) +T(n-k 2 -3)+T(n-k 1 -2) COCOA ’12, Banff August 06,

a b c d k1k1 k2k2 Counting Hamiltonian Paths a b c d k1k1 k2k2 T(n) = T(n-k 2 -2) +T(n-k 2 -3)+T(n-k 1 -2)+T(n-k 1 -3) COCOA ’12, Banff August 06,

a b c d k1k1 Counting Hamiltonian Paths a b c d k1k1 k2k2 T(n) = T(n-1)+ T(n-3) COCOA ’12, Banff August 06,

Counting Hamiltonian Paths T(n) = T(k 2 +2)+ T(k 1 +2) a b c k1k1 k2k2 a b c k1k1 k2k2 COCOA ’12, Banff August 06,

Counting Hamiltonian Paths T(n) = max{T(n-k 2 -2) + T(n-k 2 -3) + T(n-k 1 -2) + T(n-k 1 -3), T(n-1) + T(n-3), T(k 2 +2) + T(k 1 +2) } = T(n-1) + T(n-3) = O( n ) COCOA ’12, Banff August 06,

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

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,

G2G2 Child Swap Operation c u v a b c’ a’ b’ G1G1 G3G3 Let x be an ace vertex and let x be in G 1. Child swap operation does not decrease the number of Hamiltonian paths starting from x. COCOA ’12, Banff August 06,

Repeated Ancestry a b cd e x u v w y COCOA ’12, Banff August 06, v is the left child of u w is the left child of v

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

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

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

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

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

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

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

Extending Perfect Matching to a Cycle We start with a perfect matching and complete it into a cycle. b a c d e f g i j k Starting from one edge, P(n,f) = O(2 n/2 ) For O(n) edges, the number of Hamiltonian cycles is ≤ O(n2 n/2 ) COCOA ’12, Banff August 06, {a,b} {a,b,k,j}{a,b,i,e}{a,b,j,k}...

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

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

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,

Thank You