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

Dominant regions in noncrystallographic hyperplane arrangements

For a crystallographic root system, dominant regions in the Catalan hyperplane arrangement are in bijection with antichains in a partial order on the positive roots. For a noncrystallographic root system, the analogous arrangement and regions have importance in the representation theory of an associated graded Hecke algebra. Since there is also an analogous root order, it is natural to hope that a similar bijection can be used to understand these regions. We show that such a bijection does hold for type $H_3$ and for type $I_2(m)$, including arbitrary ratio of root lengths when $m$ is even, but does not hold for type $H_4$. We give a criterion that explains this failure and a list of the 16 antichains in the $H_4$ root order which correspond to empty regions.Comment: 29 pages, 5 figure

Two-sided Cayley Graphs

Cayley graphs were introduced by Arthur Cayley in 1878 to geometrically describe the algebraic structure of a group. Due to their strong symmetry, Cayley graphs find application in molecular biology, physics, and computer science. In particular, they are used in modelling interconnection networks in parallel computing. We study a new class of graphs recently introduced by Iradmusa and Praeger called two-sided Cayley graphs. Since two-sided Cayley graphs are more general and can exhibit similar symmetry, they are likely to find similar applications. We present an overview of our ongoing research which includes results in connectivity and vertex-transitivity of two-sided Cayley graphs. Pictorial examples are included to illustrate the central ideas behind our findings