John P. McSorley

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A total labeling of a graph with v vertices and e edges is defined as a one-to-one map taking the vertices and edges onto the integers 1, 2, · · · , v+e. Such a labeling is vertex magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and edge magic if the sum of an edge label and(More)
A vertex-magic total labeling of a graph G(V; E) is a one-to-one map from E ∪V onto the integers {1; 2; : : : ; |E|+ |V |} such that (x) + ∑ (xy); where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation(More)
For a simple graph G let NG(u) be the (open) neighborhood of vertex u ∈ V (G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v ∈ V (G)\{u} with NG(u) ⊆ NG(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if NH(u) 6= NH(v) when u 6= v, for all u and v ∈ V (H). In Porter and Yucas [3] a(More)
A vertex|matching-partition (V |M) of a simple graph G is a spanning collection of vertices and independent edges of G. Let vertex v ∈ V have weight wv and edge e ∈ M have weight we. Then the weight of V |M is w(V |M) = ∏ v∈V wv · ∏ e∈M we. Define the vertex|matching-partition function of G as W(G) = ∑ V |M w(V |M). In this paper we study this function when(More)
We give a construction to obtain a t-design from a t-wise balanced design. More precisely, given a positive integer k and a t(v, {k1, k2, . . . , ks}, λ) design D, with with all block-sizes ki occurring in D and 1 ≤ t ≤ k ≤ k1 < k2 < · · · < ks, the construction produces a t-(v, k, nλ) design D∗, with n = lcm( ( k1−t k−t ) , . . . , ( ks−t k−t ) ). We prove(More)
In Theorem 6.1 of [3] it was shown that, when v = r+ c− 1, every triple array TA(v, k, λrr, λcc, k : r × c) is a balanced grid BG(v, k, k : r×c). Here we prove the converse of this Theorem. Our final result is: Let v = r+ c− 1. Then every triple array is a TA(v, k, c− k, r− k, k : r × c) and every balanced grid is a BG(v, k, k : r × c), and they are(More)
I present the Triangularization Lemma which says that let P be a set of properties, each of which is inherited by quotients. If every collection of transformations on a space of dimension greater than 1 that satisfies P is reducible, then every collection of transformations satisfying P is triangularizable. I also present Burnside's Theorem which says that(More)
A single-change circular covering design (scccd) based on the set [v] = { 1 . . . . . t:} with block size k is an ordered collection o f b blocks, ~ = {Bi . . . . . B/,}, each B~ C [v], which obey: ( 1 ) each block differs from the previous block by a single element, as does the last from the first, and, (2) every pair of [v] is covered by some block. The(More)