Geoffrey Exoo

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A (k, g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions. the electronic journal of combinatorics (2013),(More)
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 graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n. We show that, for each n, al l sufficiently Igrge Paley graphs satisfy Axiom n. From this we conclude a t once that several properties of graphs are not first order, including self-complementarity and regularity. A(More)
We construct a graph of order 384, the smallest known trivalent graph of girth 14. AMS Subject Classifications: 05D25, 05D35 In this note we use a construction technique that can be viewed as a kind of generalized Cayley graph. The vertex set V of such a graph consists of the elements in multiple copies of some finite group G. The action of G on V is(More)