4 research outputs found
Distinct Distances in Graph Drawings
The \emph{distance-number} of a graph is the minimum number of distinct
edge-lengths over all straight-line drawings of in the plane. This
definition generalises many well-known concepts in combinatorial geometry. We
consider the distance-number of trees, graphs with no -minor, complete
bipartite graphs, complete graphs, and cartesian products. Our main results
concern the distance-number of graphs with bounded degree. We prove that
-vertex graphs with bounded maximum degree and bounded treewidth have
distance-number in . To conclude such a logarithmic upper
bound, both the degree and the treewidth need to be bounded. In particular, we
construct graphs with treewidth 2 and polynomial distance-number. Similarly, we
prove that there exist graphs with maximum degree 5 and arbitrarily large
distance-number. Moreover, as increases the existential lower bound on
the distance-number of -regular graphs tends to
Infinite quantum permutations
We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups.
We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting