Every longest circuit of a 3-connected, K3,3K_{3,3}-minor free graph has a chord

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no $K_{3,3}$ minor, and consequently for planar graphs.Comment: accepted by Journal of Graph Theor

Tonal prisms: iterated quantization in chromatic tonality and Ravel's 'Ondine'

The mathematics of second-order maximal evenness has far-reaching potential for application in music analysis. One of its assets is its foundation in an inherently continuous conception of pitch, a feature it shares with voice-leading geometries. This paper reformulates second-order maximal evenness as iterated quantization in voice-leading spaces, discusses the implications of viewing diatonic triads as second-order maximally even sets for the understanding of nineteenth-century modulatory schemes, and applies a second-order maximally even derivation of acoustic collections in an in-depth analysis of Ravel's ‘Ondine’. In the interaction between these two very different applications, the paper generalizes the concepts and analytical methods associated with iterated quantization and also pursues a broader argument about the mutual dependence of mathematical music theory and music analysis.Accepted manuscrip

Cluster algebras of type D: pseudotriangulations approach

We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small disk in the centre. This model provides convenient and uniform interpretations for clusters, cluster variables and their exchange relations, as well as for quivers and their mutations. We also present a new combinatorial interpretation of cluster variables in terms of perfect matchings of a graph after deleting two of its vertices. This interpretation differs from known interpretations in the literature. Its main feature, in contrast with other interpretations, is that for a fixed initial cluster seed, one or two graphs serve for the computation of all cluster variables. Finally, we discuss applications of our model to polytopal realizations of type $D$ associahedra and connections to subword complexes and $c$-cluster complexes.Comment: 21 pages, 21 figure