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2 Discussiones Mathematicae Graph Theory

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Frucht’s Theorem for the Digraph Factorial

100%

Hammack R. H.

nr 2

329-336

To every graph (or digraph) A, there is an associated automorphism group Aut(A). Frucht’s theorem asserts the converse association; that for any finite group G there is a graph (or digraph) A for which Aut(A) ∼= G. A new operation on digraphs was introduced recently as an aid in solving certain questions regarding cancellation over the direct product of digraphs. Given a digraph A, its factorial A! is certain digraph whose vertex set is the permutations of V (A). The arc set E(A!) forms a group, and the loops form a subgroup that is isomorphic to Aut(A). (So E(A!) can be regarded as an extension of Aut(A).) This note proves an analogue of Frucht’s theorem in which Aut(A) is replaced by the group E(A!). Given any finite group G, we show that there is a graph A for which E(A!) ∼= G.

Proper Connection Of Direct Products

63%

Hammack R. H. , Taylor D. T.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 4

1005-1013

The proper connection number of a graph is the least integer k for which the graph has an edge coloring with k colors, with the property that any two vertices are joined by a properly colored path. We prove that given two connected non-bipartite graphs, one of which is (vertex) 2-connected, the proper connection number of their direct product is 2.