Martin Lewinter

Hyper-Hamilton Laceable and Caterpillar-Spannable Product Graphs

Abstract We define a Hamilton laceable graph G to be hyper-Hamilton laceable (hyper HL) if either 1. (a) G is equitable, and if v is any node of G , then G − v is HL, or 2. (b) G is nearly equitable, and if v is any node in its larger color set, then G - v is HL. In particular, the hypercube Q n is hyper HL. A graph G is caterpillar-spannable (CS) if it has a spanning tree which is a caterpillar. We present several theorems concerning products of CS graphs. It is shown that the product of two CS graphs such that at least one of them has maximum degree 3 is CS.

Spanning subgraphs of a hypercube iii: meshes

An n-mesh is the cartesian product of n paths. In a binary mesh the order of each of these paths is a power of 2. We show that a mesh spans a hypercube if and only if it is abinary mesh.

The starlike trees which span a hypercube

Abstract A connected bipartite graph is called equitable if it has the same number of nodes in each of its two colors. A starlike tree with b branches is a subdivision of the star K l , b with b ⩾ 3. We prove that a starlike tree T with b branches, where 3 ⩽ b ⩽ n , having 2 n nodes spans the hypercube Q n if and only if T is equitable. This extends a result of Nebesky who had made this observation for b = 3.

Spanning subgraphs of a hypercube II: Double starlike trees

A double star is a tree with exactly two nodes which are not endnodes. A double starlike tree T is a subdivision of a double star S in which the edge joining the two central nodes of S is not subdivided. We characterize the double starlike trees with maximum degree three that span a hypercube.

Which Double Starlike Trees Span Ladders

Abstract The hypercube Q n is defined recursively by Q 1 = K 2 and Q n = Q n-1 x K 2 .| V ( Q n )| = 2 n . A ladder on 2 k vertices is K 2 × P K . Ladders on 2 n vertices span Q n . A double starlike tree is a tree with exactly two vertices of degree greater than two. We show which double starlike trees span ladders, implying that such double starlike trees on 2 n vertices span Q n .

Graphs with Special Distance Properties

Abstract A graph G is called an F-graph if its center contains at least two vertices, and if x and y are in the center, then the distance between x and y is equal to the radius of G. The existence of such graphs is surprising and is in marked contrast to trees whose center vertices must be adjacent. Some properties of F-graphs are presented, along with open questions. Recent results on other graphs with unusual distance properties are included.