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The Heawood graph and K 3;3 have the property that all of their 2-factors are Hamilton circuits. We call such graphs 2-factor hamiltonian. We prove that if G is a k-regular bipartite 2-factor hamiltonian graph then either G is a circuit or k ¼ 3: Furthermore, we construct an infinite family of cubic bipartite 2-factor hamiltonian graphs based on the Heawood… (More)

The Heawood graph and K 3,3 have the property that all of their 2–factors are hamiltonian cycles. We call such graphs 2–factor hamiltonian. More generally, we say that a connected k–regular bipartite graph G belongs to the class BU(k) if for each pair of 2-factors, F 1 , F 2 in G, F 1 and F 2 are isomorphic. We prove that if G ∈ BU(k) , then either G is a… (More)

Let G be a connected k–regular bipartite graph with bipartition V (G) = X ∪Y and adjacency matrix A. We say G is det–extremal if per(A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det–extremal 3–connected cubic bipartite graphs. We extend McCuaig's result by determining the structure of… (More)

We show that a digraph which contains a directed 2-factor and has minimum in-degree and out-degree at least four has two non-isomorphic directed 2-factors. As a corollary we deduce that every graph which contains a 2-factor and has minimum degree at least eight has two non-isomorphic 2-factors. In addition we construct: an infinite family of strongly… (More)

Configurations of type (κ 2 + 1)κ give rise to κ–regular simple graphs via configuration graphs. On the other hand, neighbourhood geometries of C4–free κ–regular simple graphs on κ 2 + 1 vertices turn out to be configurations of type (κ 2 + 1)κ. We investigate which configurations of type (κ 2 + 1)κ are equal or isomorphic to the neighbourhood geometry of… (More)