aequationes mathematicae

, Volume 23, Issue 1, pp 80–97

# Enumeration of hamiltonian paths in Cayley diagrams

• David Housman
Research Paper

## Abstract

LetG be a group generated by a subset of elementsS. The Cayley diagram ofG givenS is the labeled directed graph with vertices identified with the elements ofG and (v, u) is an edge labeledh ifhS anduh=v. The sequence of elements ofS corresponding to the edges transversed in a hamiltonian path (whose initial vertex is the identity) is called a group generating sequence (abbreviatedggs) inS.

In this paper a minimal upper bound for the number ofggs's in a pair of generator elements for any two-generated group is given. For all groups of the formG=〈a, b:b n =1,a m =b r ,ba=ab−1〉 wherem is even, it is shown that the number ofggs's in {a, b} is 1+m(n−1)/2. An algorithm is developed that yields the number ofggs's for two-generated groupsG=〈a, b〉 for which 〈ba−1〉⊲G. Explicit forms for the countedggs's are also provided.

## AMS (1980) subject classification

Primary 05C25 Secondary 05C30

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