Oriented Flip Graphs and Noncrossing Tree Partitions

International audience Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.

Cluster Automorphisms and the Marked Exchange Graphs of Skew-Symmetrizable Cluster Algebras

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.

A Cambrian Framework for the Oriented Cycle

This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and $\mathbf{g}$-vector fan associated to any exchange matrix $B$ whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically oriented $n$-cycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for $B$ with an antipodal copy of the Cambrian fan for $-B$. In this paper, we extend this "doubled Cambrian fan'' construction to the oriented $n$-cycle, using a more general notion of sortable elements for quivers with cycles.

Linear independence of cluster monomials for skew-symmetric cluster algebras

Fomin–Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.

AN APPLICATION OF ST-NUMBERING TO SECRET KEY AGREEMENT

Assume that there are players and an eavesdropper Eve, where several pairs of players have shared secret keys beforehand. We regard each player as a vertex of a graph and regard each pair of players sharing a key as an edge. Consider the case where Eve knows some of the keys according to a certain probability distribution. In this paper, applying the technique of st-numbering, we propose a protocol which allows any two designated players to agree on a secret key through such a "partially leaked key exchange graph." Our protocol is optimal in the sense that Eve's knowledge about the secret key agreed on by the two players is as small as possible.