83 research outputs found
Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups
Generalizing a result of Conway, Sloane, and Wilkes for real reflection
groups, we show the Cayley graph of an imprimitive complex reflection group
with respect to standard generating reflections has a Hamiltonian cycle. This
is consistent with the long-standing conjecture that for every finite group, G,
and every set of generators, S, of G the undirected Cayley graph of G with
respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments,
to appear in Discrete Mathematic
2-generated Cayley digraphs on nilpotent groups have hamiltonian paths
Suppose G is a nilpotent, finite group. We show that if {a,b} is any
2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b)
has a hamiltonian path. This implies there is a hamiltonian path in every
connected Cayley graph on G that has valence at most 4.Comment: 7 pages, no figures; corrected a few typographical error
On Cayley digraphs that do not have hamiltonian paths
We construct an infinite family of connected, 2-generated Cayley digraphs
Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the
generators a and b are arbitrarily large. We also prove that if G is any finite
group with |[G,G]| < 4, then every connected Cayley digraph on G has a
hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or
5).Comment: 10 pages, plus 14-page appendix of notes to aid the refere
Groups all of whose undirected Cayley graphs are integral
Let be a finite group, be a set such that if
, then , where denotes the identity element of .
The undirected Cayley graph of over the set is the graph
whose vertex set is and two vertices and are adjacent whenever
. The adjacency spectrum of a graph is the multiset of all
eigenvalues of the adjacency matrix of the graph. A graph is called integral
whenever all adjacency spectrum elements are integers. Following Klotz and
Sander, we call a group Cayley integral whenever all undirected Cayley
graphs over are integral. Finite abelian Cayley integral groups are
classified by Klotz and Sander as finite abelian groups of exponent dividing
or . Klotz and Sander have proposed the determination of all non-abelian
Cayley integral groups. In this paper we complete the classification of finite
Cayley integral groups by proving that finite non-abelian Cayley integral
groups are the symmetric group of degree , and
for some integer , where is the
quaternion group of order .Comment: Title is change
Cayley graphs of order 27p are hamiltonian
Suppose G is a finite group, such that |G| = 27p, where p is prime. We show
that if S is any generating set of G, then there is a hamiltonian cycle in the
corresponding Cayley graph Cay(G;S).Comment: 13 pages, no figures; minor revisions, including suggestions from a
referee; this version is to appear in the International Journal of
Combinatoric
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