This paper addresses the existence of Hamiltonian paths and cycles in two-dimensional grids consisting of triangles or quadrilaterals, and three-dimensional grids consisting of tetrahedra or hexahedra. The paths and cycles may be constrained to pass from one element to the next through an edge, through a vertex, or be unconstrained and pass through either. It was previously known that an uncons...

A path π = (v1, v2, . . . , vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(vi+1), where deg(vi) denotes the degree of vertex vi ∈ V. The downhill domination number equals the minimum cardinality of a set S ⊆ V having the property that every vertex v ∈ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers o...

Kreweras’ conjecture [1] asserts that every perfect matching of the hypercube Qd can be extended to a Hamiltonian cycle. We [2] proved this conjecture but here we present a simplified proof. The matching graph M(G) of a graph G has a vertex set of all perfect matchings of G, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We ...

In 2007, in their paper Path coverings with prescribed ends in faulty hypercubes, N. Castañeda and I. Gotchev formulated the following conjecture: Let n and k be positive integers with n ≥ k + 3 and F be a set of k even (odd) and k + 1 odd (even) vertices in the binary hypercube Qn. If u1 and u2 are two distinct even (odd) vertices inQn − F then forQn − F there exists a Hamiltonian path that co...

The mirror (or bipartite complement) mir(B) of a bipartite graph B = (X,Y,E) has the same color classes X and Y as B, and two vertices x ∈ X and y ∈ Y are adjacent in mir(B) if and only if xy / ∈ E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal ...

In 2007, in their paper Path coverings with prescribed ends in faulty hypercubes, N. Castañeda and I. Gotchev formulated the following conjecture: Let n and k be positive integers with n ≥ k + 3 and F be a set of k even (odd) and k + 1 odd (even) vertices in the binary hypercube Qn. If u1 and u2 are two distinct even (odd) vertices inQn − F then forQn − F there exists a Hamiltonian path that co...

Let Pn be a set of n m points that are the vertices of a convex polygon and let Mm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross and edges joining two perfect matchings M and M if M M a b c d a d b c for some points a b c d of Pn We prove the following results about Mm its diameter is m it is bipartite for e...

Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgraph of G whose components are nontrivial paths. Let k be the numbe...

We introduce a new property for geometric graphs: the local diameter. This property is based on the use of the region counting distances introduced by Demaine, Iacono and Langerman as a generalization of the rank difference. We use it as a characterization of the local density of the vertices. We study the properties of those distances, and show that there is a strong relation between region co...

The Kneser graph K(n, k) has as vertices all k-element subsets of [n] := {1, 2, . . . , n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n, k) has as vertices all k-element and (n−k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all connected Kneser graphs and bipar...