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Sudip Biswas1, Stephane Durocher2,Debajyoti Mondal2 and Rahnuma Islam Nishat3Hamiltonian Paths and Cycles in Planar Graphs1Department of Computer Science, Louisiana State University, USA2Department of Computer Science, University of Manitoba, Canada3Department of Computer Science, University of Victoria, CanadaProblem Definition COCOA 12, Banff August 06, 2012Planar Graph2Problem DefinitionPlanar GraphO(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA 12, Banff August 06, 20123Problem DefinitionOuterplanar GraphO(n1.46557n) upper bound and (1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph COCOA 12, Banff August 06, 20124Previous WorksEric T. Bax (1993)O(2nn4) time and O(n3log 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 mn grid graphs for m=1,2,3,4,5.Eppstein (2004)Count all Hamiltonian cycles in a degree three graph in time O(23n/8) 1.297nGebauer (2011)O(1.783n) upper bound on the number of Hamiltonian cycles in 4-regular graphs COCOA 12, Banff August 06, 20125Previous WorksBuchin et al. (2007)de Mier and Noy(2009)O(2.3404n) upper bound and (2.0845n) lower bound on the number of Hamiltonian cycles in planar graphs.O(2.8927n) upper bound and a (2.4262n) lower bound on the number of simple cycles in planar graphs.(1.502837 n) bound on the number of simple cycles in outerplanar graphs. COCOA 12, Banff August 06, 20126Our ResultsO(n1.46557n) upper bound and (1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graphFor any positive integer n 6, we dene 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(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA 12, Banff August 06, 20127Counting Hamiltonian Pathsabcdk1k2abcdk1k2T(n) =T(n-k2-2) COCOA 12, Banff August 06, 20128abcdk1k2Counting Hamiltonian Pathsabcdk1k2T(n) = T(n-k2-2)+T(n-k2-3) COCOA 12, Banff August 06, 20129abcdk1k2Counting Hamiltonian Pathsabcdk1k2T(n) = T(n-k2-2)+T(n-k2-3)+T(n-k1-2) COCOA 12, Banff August 06, 201210abcdk1k2Counting Hamiltonian Pathsabcdk1k2T(n) = T(n-k2-2)+T(n-k2-3)+T(n-k1-2)+T(n-k1-3) COCOA 12, Banff August 06, 201211abcdk1Counting Hamiltonian Pathsabcdk1k2T(n) = T(n-1)+ T(n-3) COCOA 12, Banff August 06, 201212Counting Hamiltonian PathsT(n) = T(k2+2)+ T(k1+2)abck1k2abck1k2 COCOA 12, Banff August 06, 201213Counting Hamiltonian PathsT(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, 201214Our ResultsO(n1.46557n) upper bound and (1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graphFor any positive integer n 6, we dene 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(2.2134n) 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, 201215Maximum Hamiltonian Paths Starting from a VertexOuterplanar Graphs with 7 verticesStep 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, 201216G2Child Swap OperationcuvabcabG1G3Let x be an ace vertex and let x be in G1.

Child swap operation does not decrease the number of Hamiltonian paths starting from x. COCOA 12, Banff August 06, 201217Repeated Ancestry abcdexuvwy COCOA 12, Banff August 06, 201218v is the left child of uw is the left child of v

Child Flip and Parent Flipv is the left child of uw is the left child of v

abdcexuvwyIf ace vertex is not e apply child flipabcdeabcuxvwyzOtherwise apply parent flip COCOA 12, Banff August 06, 201219abcdexuvwyZigZag GraphazbyAce vertexT(n) = T(n-1) + T(n-3)O(n1.46557n) upper bound and (1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graph COCOA 12, Banff August 06, 20122020Our ResultsO(n1.46557n) upper bound and (1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graphFor any positive integer n 6, we dene 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(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA 12, Banff August 06, 201221Cycle-Paths in Planar GraphsA cycle-path is a simple path that can be extended to a simple cycle.bacdefgijkSplit the edges incident to each vertex into two sets COCOA 12, Banff August 06, 201222Cycle-Paths in Planar GraphsA cycle-path is a simple path that can be extended to a simple cycle.bacdefgijk{a}{a,b}{a,d} COCOA 12, Banff August 06, 201223Cycle-Paths in Planar GraphsA cycle-path is a simple path that can be extended to a simple cycle.bacdefgijk{a}{a,b}{a,d}{a,b,i}{a,b,k}{a,b,j} COCOA 12, Banff August 06, 201224Cycle-Paths in Planar GraphsA cycle-path is a simple path that can be extended to a simple cycle.bacdefgijk{a}{a,b}{a,d}{a,b,i}{a,b,k}{a,b,j}. . .. . .. . .P(n,f) kv.P(n-1,f-kv+1) +1 P(n,f) = O(2n)

COCOA 12, Banff August 06, 201225Extending Perfect Matching to a CycleWe start with a perfect matching and complete it into a cycle.bacdefgijkStarting from one edge, P(n,f) = O(2n/2)For O(n) edges, the number of Hamiltonian cycles is O(n2n/2)

COCOA 12, Banff August 06, 201226{a,b}{a,b,k,j}{a,b,i,e}{a,b,j,k}. . .. . .. . .Upper Bound on the Number of Hamiltonian CyclesEach 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 is6n/4 O(n). 2n/2 < O(n).2.2134n COCOA 12, Banff August 06, 201227Summary of Our ResultsO(n1.46557n) upper bound and (1.46557n) lower bound on the number of Hamiltonian paths in an outerplanar graphFor any positive integer n 6, we dene 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(2.2134n) upper bound on the number of Hamiltonian cycles in any planar graph. COCOA 12, Banff August 06, 201228Open ProblemsCan 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, 201229Thank You