835 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
Recent trends and future directions in vertex-transitive graphs
A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade
An update on the middle levels problem
The middle levels problem is to find a Hamilton cycle in the middle levels,
M_{2k+1}, of the Hasse diagram of B_{2k+1} (the partially ordered set of
subsets of a 2k+1-element set ordered by inclusion). Previously, the best
result was that M_{2k+1} is Hamiltonian for all positive k through k=15. In
this note we announce that M_{33} and M_{35} have Hamilton cycles. The result
was achieved by an algorithmic improvement that made it possible to find a
Hamilton path in a reduced graph of complementary necklace pairs having
129,644,790 vertices, using a 64-bit personal computer.Comment: 11 pages, 5 figure
Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph
The Bubble-sort graph , is a Cayley graph over the
symmetric group generated by transpositions from the set . It is a bipartite graph containing all even cycles of
length , where . We give an explicit
combinatorial characterization of all its - and -cycles. Based on this
characterization, we define generalized prisms in , and
present a new approach to construct a Hamiltonian cycle based on these
generalized prisms.Comment: 13 pages, 7 figure
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