Enumerating Hamiltonian Cycles

Enumerating Hamiltonian Cycles

A dynamic programming method for enumerating hamiltonian cycles in arbitrary graphs is presented. The method is applied to grid graphs, king’s graphs, triangular grids, and three-dimensional grid graphs, and results are obtained for larger cases than previously published. The approach can easily be modified to enumerate hamiltonian paths and other similar structures.

Enumerating and Counting Cycles in Bipartite Graphs

On Hamiltonian cycles and Hamiltonian paths

A Hamiltonian cycle is a spanning cycle in a graph, i.e., a cycle through every vertex, and a Hamiltonian path is a spanning path. In this paper we present two theorems stating sufficient conditions for a graph to possess Hamiltonian cycles and Hamiltonian paths. The significance of the theorems is discussed, and it is shown that the famous Ore’s theorem directly follows from our result.  2004...

Finding Hidden Hamiltonian Cycles

Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that “hide” the cycle. Is it possible to unravel the structure, that is, to efficiently find a Hamiltonian cycle in G? We describe an O(n3 logn) steps algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the “trouble spots” of G by a collection of ...

Mutually Independent Hamiltonian Cycles

A Hamiltonian cycle of a graph G is a cycle which contains all vertices of G. Two Hamiltonian cycles C1 = 〈u0, u1, u2, ..., un−1, u0〉 and C2 = 〈v0, v1, v2, ..., vn−1, v0〉 in G are independent if u0 = v0, ui 6= vi for all 1 ≤ i ≤ n − 1. If any two Hamiltonian cycles of a Hamiltonian cycles set C = {C1, C2, ..., Ck} are independent, we call C is mutually independent. The mutually independent Hami...