Derek A. Holton

On defect-d matchings in graphs

A defect-d matching in a graph G is a matching which covers all but d vertices of G. G is d-covered if each edge of G belongs to a defect-d matching. Here we characterise d-covered graphs and d-covered connected bipartite graphs. We show that a regular graph G of degree r which is (r � 1)-edge-connected is 0-covered or 1-covered depending on whether G has an even or odd number of vertices, but, given any non-negative integers r and d, there exists a graph regular of degree r with connectivity and edge-connectivity r � 2 which does not even have a defect-d matching. Finally, we prove that a vertex-transitive graph is 0-covered or 1-covered depending on whether it has an even or odd number of vertices.

A nine point theorem for 3-connected graphs

We prove that a 3-connected cubic graph contains a cycle through any nine points.

Hamiltonian cycles in cubic 3-connected bipartite planar graphs

Abstract We show that 3-connected cubic bipartite planar graphs with fewer than 66 vertices are Hamiltonian.

Rings of bonds in graphs

Abstract It is shown that every strict maximal ring of bonds of size greater than 3 is even.

REGULAR GRAPHS AND STABILITY

We show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gy, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if F(G) contains a transposition, offering some evidence for its truth.

Two applications of semi-stable graphs

A Michigan graph G on a vertex set V is called semi-stable if for some @u@eV, @C(G@u) = @C(G)@u. It can be shown that all regular graphs are semi-stable and this fact is used to show (i) that if @C(G) is doubly transitive then G = Kn or K@?n, and (ii) that @C(G) can be recovered from @C(G@u). The second result is extended to the case of stable graphs.

Cycles through ten vertices in 3-connected cubic graphs

It is known that there exists a cycle through any nine vertices of a 3-connected cubic graphG. Here we show that if an edge is removed from such a graph, then there is still a cycle through any five vertices. Furthermore, we characterise the circumstances in which there fails to be a cycle through six. As corollaries we are able to prove that a 3-connected cubic graph has a cycle through any specified five vertices and one edge, and to classify the conditions under which it has a cycle through four chosen vertices and two edges.We are able to use the five and six vertex results to show that a 3-connected cubic graph has a cycle which passes through any ten given vertices if and only if the graph is not contractible to the Petersen graph in such a way that the ten vertices each map to a distinct vertex of the Petersen graph.

On the automorphism group of a tree

It is shown that H = Γ(T)v is normal in G = Γ(Tv) for any tree T and any vertex v, if and only if, for all vertices u in the neighborhood N of v, the set of images of u under G is either contained in N or has precisely the vertex u in common with N and every vertex in the set of images is fixed by H. Further, if S is the smallest normal subgroup of G containing H then GS is the direct product of the wreath products of various symmetric groups around groups of order 1 or 2. The degrees of the symmetric groups involved depend on the numbers of isomorphic components of Tv and the structure of such components.

Stability of cartesian products

Abstract We complete the work started by Holton and Grant concerning the semi-stability of non-trivial connected cartesian products and show that all such products are semi-stable. Further we show that except for certain (listed) restricted graphs, connected cartesian products are semi-stable at every vertex. Finally, we show that the cartesian product of any two graphs is not semi-stable if and only if one of them is totally disconnected and the other is not semi-stable.

Cycles through four edges in 3-connected cubic graphs

We give necessary and sufficient conditions for four edges in a 3-connected cubic graph to lie on a cycle. As a consequence, if such a graph is cyclically 4-edge-connected with order greater than 8 it is shown that any four independent edges lie on a cycle.