Hamilton Cycles with Require Almost Perfect Patchings in Hypercubes Cheng-Kuan Lin Jimmy J. M. Tan Lih-Hsing Hsu

Hamiltonian Cycle A cycle is a hamiltonian cycle of graph G if it contains all the vertices of G.

Matching A set of edges P ⊂ E of a graph G = (V ,E) is a matching if every vertex of G is incident with at most one edge of P . If a vertex v of G is incident with an edge of P, we say that v is covered by P . A matching P is perfect if every vertex of G is covered by P.

Kreweras’ conjecture Every perfect matching of the hypercube extends to a Hamiltonian cycle. G. Kreweras, Matchings and Hamiltonian cycles on hypercubes, Bull. Inst. Combin. Appl. 16 (1996) 87–91.

Fink’s Theorem For every perfect matching P of KQn there exists a perfect matching R of Qn , n  2, such that P ∪ R is a Hamilton cycle of KQn. J. Fink, Perfect matchings extend to Hamilton cycles in hypercubes, Journal of Combinatorial Theory, Series B 97 (2007) 1074–1076.

R. Škrekovski’s conjecture Every matching of n-dimensional hypercube with n  2 can be extended to a Hamilton cycle. J. Fink, Perfect matchings extend to Hamilton cycles in hypercubes, Journal of Combinatorial Theory, Series B 97 (2007) 1074–1076.

Gregor’s Result Let F be a subcubic partition of a set A  V(Qn) where n  1, and let P be a perfect matching of K[A]. Then, there exists a perfect matching R of Qn[A] − F such that P∪R forms a Hamiltonian cycle of K[A] if and only if Qn[A] − F + P is connected. Petr Gregor, Perfect matchings extending on subcubes to Hamiltonian cycles of hypercubes, Discrete Mathematics, In Press, Corrected Proof, Available online 24 March 2008.

Gregor’s Result Q100 F

Gregor’s Result Q100 Q3 Q10 F Q5 Q5 Q20 Q8

Gregor’s Result Q100 Q3 Q10 P F Q5 Q5 Q20 Q8

Gregor’s Result Q100 Q3 Q10 P F Q5 Q5 Q20 Q8 There is a perfect matching R of Qn[A] − F such that P∪R forms a Hamiltonian cycle of K[A]. Q3 Q10 P F Q5 Q5 Q20 Q8

Perfect matchings can not extend to hamiltonian laceable

Perfect matchings can not extend to hamiltonian laceable

Perfect matchings can not extend to hamiltonian laceable u v 1. u and v can not adjacent in the matching.

Perfect matchings can not extend to hamiltonian laceable u 1. u and v can not adjacent in the matching. 2. there is a perfect matching such that it can be extend to a hamiltonian path between u and v. v

Perfect matchings can not extend to hamiltonian laceable u Is almost perfect matching can be extend to a hamiltonian path between u and v? v

Perfect matchings can not extend to hamiltonian laceable u Is almost perfect matching can be extend to a hamiltonian path between u and v? v

Theorem Let u and v be any two vertices in distinct partite sets of KQn-1,Qn-1. For any almost perfect matching P of KQn-1,Qn-1 such that either u or v is not covered by P, there is a perfect R of Q such that P ∪ R is a Hamilton path of KQn-1,Qn-1 between u and v.

Theorem For every almost perfect matching P of KQn-1, Qn-1, there is a edge subset R of Qn with |R| = 2n-1 + 1 such that P ∪ R forms a Hamilton cycle.

Conjecture (n  4?) Let u and v be any two vertices of distinct partite sets of Qn. For any non-perfect matching P of Qn with (u, v)  P, P can be extended to a Hamilton path of Qn between u and v.

References [1] T. Dvořák, Hamiltonian cycles with prescribed edges in hypercubes, SIAM J. Discrete Math. 19 (2005) 135–144. [2] D. Dimitrov, R. Škrekovski, T. Dvořák, P. Gregor, Gray Codes Faulting Matchings, manuscript. [3] J. Fink, Perfect matchings extend to Hamilton cycles in hypercubes, Journal of Combinatorial Theory, Series B 97 (2007) 1074–1076. [4] L. Gros, Théorie du Baguenodier, Aimé Vingtrinier, Lyon, 1872. [5] P. Gregor, Perfect matchings extending on subcubes to Hamiltonian cycles of hypercubes, Discrete Mathematics, In Press, Corrected Proof, Available online 24 March 2008. [6] G. Kreweras, Matchings and Hamiltonian cycles on hypercubes, Bull. Inst. Combin. Appl. 16 (1996) 87–91. [7] C. Savage, A survey of combinatorial Gray codes, SIAM Rev. 39 (1997) 605–629.