Michael Edwin Raines

The Decomposition Dimension of Graphs

Abstract. For an ordered k-decomposition ? = {G1, G2,…,Gk} of a connected graph G and an edge e of G, the ?-representation of e is the k-tuple r(e|?) = (d(e, G1), d(e, G2),…,d(e, Gk)), where d(e, Gi) is the distance from e to Gi. A decomposition ? is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n≥ 2, there exists a tree T of order n with dec(T) = k. It is also shown that dec(G) ≤n for every graph G of order n≥ 3 and that dec(Kn) ≤⌊(2n + 5)/3⌋ for n≥ 3.

Embedding extended Mendelsohn triple systems

Abstract In this paper, it is shown that every extended Mendelsohn triple system of order n and index λ can be embedded in an extended Mendelsohn triple system of order v > n and index λ if and only if v ⩾2 n .

A generalization of the Doyen-Wilson theorem for extended triple systems of all indices

Abstract In this paper, it is shown that every extended triple system of order n and index λ > 1 can be embedded in an extended triple system of order v and index λ > 1 if and only if v = n or v ⩾ 2 n . This extends results obtained by Hoffman and Rodger when λ = 1 and gives a generalization of the Doyen-Wilson theorem for extended triple systems of all indices.

Orientation distance graphs

For two nonisomorphic orientations D and D2 of a graph G, the orientation distance do(D,D2) between D and D2 is the minimum number of arcs of D whose directions must be reversed to produce an orientation isomorphic to D2. The orientation distance graph 𝒟o(G) of G has the set 𝒪(G) of pairwise nonisomorphic orientations of G as its vertex set and two vertices D and D2 of 𝒟0(G) are adjacent if and only if do(D,D2) = 1. For a nonempty subset S of 𝒪(G), the orientation distance graph 𝒟0(S) of S is the induced subgraph )S* of 𝒟o(G). A graph H is an orientation distance graph if there exists a graph G and a set S⊆ 𝒪(G) such that 𝒟o(S) is isomorphic to H. In this case, H is said to be an orientation distance graph with respect to G. This paper deals primarily with orientation distance graphs with respect to paths. For every integer n ≥4, it is shown that 𝒟o(Pn) is Hamiltonian if and only if n is even. Also, the orientation distance graph of a path of odd order is bipartite. Furthermore, every tree is an orientation distance graph with respect to some path, as is every cycle, and for n ≥ 3 the clique number of 𝒟o(Pn) is 2 if n is odd and is 3 otherwise.