T. D. Parsons

Isomorphism of circulant graphs and digraphs

Abstract Let S⊆ {1, …, n−1} satisfy −S = S mod n. The circulant graph G(n, S) with vertex set {v0, v1,…, vn−1} and edge set E satisfies vivj ϵ E if and only if j − i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = − S. Adam conjectured that G(n, S) ≊ G(n, S′) if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p2, and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime, or n is divisible by 24.

Path-cycle Ramsey numbers

Recently, Ramsey numbers have been obtained for several classes of graphs. In particular, they have been studied for graphs of low order, pairs of paths, pairs of cycles, and for a star and a path. In this paper, the Ramsey numbers are obtained for all path-cycle pairs.

Hamiltonian paths in vertex-symmetric graphs of order 4p

It is shown that every connected vertex-symmetric graph of order 4p (p a prime) has a Hamiltonian path.

On Hamiltonian Cycles in Metacirculant Graphs

In this paper it is shown that every connected metacirculant with an odd number of vertices greater than one and with blocks of prime cardinality has a hamiltonian cycle.

Hamilton Cycles in Metacirculant Graphs with Prime Cardinality Blocks

In this paper it is shown that every connected metacirculant graph having an even number of blocks of prime cardinality, other than the sole exception of the Petersen graph, has a Hamilton cycle.

A duality theorem for graph embeddings

A generalized type of graph covering, called a “Wrapped quasicovering” (wqc) is defined. If K, L are graphs dually embedded in an orientable surface S, then we may lift these embeddings to embeddings of dual graphs K,L in orientable surfaces S, such that S are branched covers of S and the restrictions of the branched coverings to K,L are wqcs of K, L. the theory is applied to obtain genus embeddings of composition graphs G[nK1] from embeddings of “quotient” graphs G.

Longest cycles in r-regular r-connected graphs

Abstract Let r ≥ 3 be an integer, and e > 0 a real number. It is shown that there is an integer N(r, e) such that for all n ≥ N (if r is even) or for all even n ≥ N (if r is odd), there is an r-connected regular graph of valency r on exactly n vertices whose longest cycles have fewer than en vertices. That is, the number e > 0, no matter how small, is a “shortness coefficient” for the family of r-valent regular r-connected graphs.

Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let Gm,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a1, …, am, each vertex b of B by an independent set b1,…, bn, and each edge ab of G by the complete bipartite graph with edges aibj (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of Gm,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.