The 35 th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing

s of invited talks Metamorphoses of graph designs Elizabeth J. Billington University of Queensland A lambda-fold graph design, or G-design of order n, is an edge-disjoint decomposition of a λ-fold complete graph, λKn, into isomorphic copies of some graph G. (Here λKn denotes the graph on n vertices with precisely λ edges between every pair of vertices.) Let H be a subgraph of G. A metamorphosis of a G-design into an H-design is obtained when the following is possible. Suppose there are b copies of G (called blocks) in the G-design, say Gi, 1 6 i 6 b. For each block Gi, we retain its subgraph Hi, isomorphic to H, and we rearrange all the “discarded” edges, {E(Gi\Hi) | 1 6 i 6 b}, into further copies of the graph H. The result is a lambda-fold H-design of order n, obtained by metamorphosis of the G-design. Metamorphoses of graph designs were first considered late in the last century. This talk will present a survey of results in this area, some open problems and new directions. (Tuesday 9:00) Combinatorial properties of transformation monoids Peter J. Cameron Queen Mary, University of London The field of permutation groups has had close links with combinatorics for nearly a century, and includes the work of Witt on Steiner systems and the Mathieu groups, Sims on graphs and permutation groups, Higman on coherent configurations, and many other topics. By contrast, the study of transformation monoids is not well developed, and potential links with combinatorics await study. A transformation monoid is synchronizing if it contains a transformation whose image has cardinality 1. Study of synchronizing monoids was initiated by the celebrated Černý conjecture, more than forty years old and still open. It is conjectured that the probability that two random transformations on n points generate a synchronizing monoid tends to 1 as n → ∞. This would be an analogue of Dixon’s theorem, asserting that the probability that two random permutations generate the symmetric or alternating group tends to 1. As in Dixon’s theorem, the proof strategy would involve describing the maximal non-synchronizing monoids (this has been done, in terms of graphs), and then doing inclusion-exclusion (this step has not been done yet). By abuse of language, a permutation group G is said to be synchronizing if the monoid 〈G, f〉 is synchronizing for any non-bijective transformation f . This condition on permutation groups implies primitivity but is strictly stronger; testing it for a specific permutation group