Edge-disjoint Hamilton cycles in 4-regular planar graphs.
Edge-disjoint Hamiltonian cycles in two-dimensional torus.
Ein planarer hypohamiltonscher Graph mit 57 Knoten.
Embeddings of hamiltonian paths in faulty k-ary 2-cubes
It is well known that the k-ary n-cube has been one of the most efficient interconnection networks for distributed-memory parallel systems. A k-ary n-cube is bipartite if and only if k is even. Let (X,Y) be a bipartition of a k-ary 2-cube (even integer k ≥ 4). In this paper, we prove that for any two healthy vertices u ∈ X, v ∈ Y, there exists a hamiltonian path from u to v in the faulty k-ary 2-cube with one faulty vertex in each part.
Enumerating all Hamilton cycles and bounding the number of Hamilton cycles in 3-regular graphs.
Enumeration of Hamiltonian circuits in rectangular grids.
Enumeration of hamiltonian paths in Caley diagrams.
Equations relating factors in decompositions into factors of some family of plane triangulations, and applications (with an appendix by Andrzej Schinzel)
Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is...
Extension of several sufficient conditions for Hamiltonian graphs
Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph , 1 ≤ r ≤ s ≤ t or the graph . It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph...
Extremal problems for forbidden pairs that imply hamiltonicity
Let C denote the claw , N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free (does...