Robert L. Hemminger

The group of an X-join of graphs*

In this paper we give necessary and sufficient conditions for the group of graph automorphisms of the X-join of {Yx}x∈X to be the “natural” ones, i.e., those that are obtained by first permuting the Yx according to a permutation of subscripts by an automorphism of X and then performing an arbitrary automorphism of each Yx. This simultaneously solves and generalizes the problem of giving necessary and sufficient conditions for the group of the lexicographic product of two graphs to be the wreath product of the groups of the factor graphs. This problem was introduced by Harary and partial solutions had been given by Sabidussi and the author.

P 3 -isomorphisms for graphs

The P3-graph of a finite simple graph G is the graph whose vertices are the 3-vertex paths of G, with adjacency between two such paths whenever their union is a 4-vertex path or a 3-cycle. In this paper we show that connected fnite simple graphs G and H with isomorphic P3-graphs are either isomorphic or part of three exceptional families. We also characterize all isomorphisms between P3-graphs in terms of the original graphs.

On 3-skein isomorphisms of graphs

It is shown that a 3-skein isomorphism between 3-connected graphs with at least 5 vertices is induced by an isomorphism. These graphs have no loops but may be infinite and have multiple edges.

The 3-connected graphs having a longest cycle containing only three contractible edges

It is shown that with one small exception, the 3-connected graphs admitting longest cycles that contain less than four contractible edges of the parent graph are the members of three closely related infinite families.

Covering contractible edges in 3-connected graphs: II. characterizing those with covers of size three

It is shown that if G is a 3-connected graph with |V(G)| ≥ 10, then, with the exception of one infinite class based on K3,p, it takes at least four vertices to cover the set of contractible edges of G.

Contractible edges in longest cycles in non-Hamiltonian graphs

Abstract An edge of a 3-connected graph is contractible if its contraction results in a graph which is still 3-connected. All 3-connected graphs with seven or more vertices are known to have at least three contractible edges on any longest cycle. Recently, it has been conjectured that any non-Hamiltonian 3-connected graph has at least six contractible edges on any longest cycle. We prove this conjecture and provide a construction to show that it is best possible.

On 3-connected graphs with contractible edge covers of size k

Abstract We consider 3-connected graphs whose contractible edges can be covered by k vertices and show that they can be divided into finitely many infinite classes of graphs.

On n-skein isomorphisms of graphs

It is shown that a 3-skein isomorphism between 3-connected graphs with at least 5 vertices is induced by an isomorphism. These graphs have no loops but may be infinite and have multiple edges.

Infinite digraphs isomorphic with their line digraphs

In this paper we give a structural characterization of the digraphs that are isomorphic with their line digraphs.

Integer sequences with proscribed differences and bounded growth rate

Let n, b and c be positive integers with b ⩽ c and let A = a i ini = 0 n be a sequence of integers such that 0 = a 0 a 1 a n and a i + b ⩽ a i + c for all i with 0 ⩽ i ⩽ n − b . We find all n function of b, c and k, k a positive integer, so that all such sequences have no two members that differ by exactly k .