Spanning hypertrees, vertex tours and meanders

TL;DR

This paper explores spanning hypertrees in hypermaps, establishing bijections with meanders and semimeanders, and introduces hyperdeletions and hypercontractions to recursively count hypertrees and connect to the Tutte polynomial.

Contribution

It generalizes the concept of spanning trees to hypermaps, introduces hyperdeletions and hypercontractions, and links these to meanders, semimeanders, and the Tutte polynomial.

Findings

Bijection between hypertrees and meanders of order n.

Recursive counting of hypertrees via hyperdeletions and hypercontractions.

Connection of hypertrees enumeration to the Tutte polynomial.

Abstract

This paper revisits the notion of a spanning hypertree of a hypermap introduced by one of its authors and shows that it allows to shed new light on a very diverse set of recent results. The tour of a map along one of its spanning trees used by Bernardi may be generalized to hypermaps and we show that it is equivalent to a dual tour described by Cori and Mach\`\i. We give a bijection between the spanning hypertrees of the reciprocal of the plane graph with $2$ vertices and $n$ parallel edges and the meanders of order $n$ and a bijection of the same kind between semimeanders of order $n$ and spanning hypertrees of the reciprocal of a plane graph with a single vertex and $n/2$ nested edges. We introduce hyperdeletions and hypercontractions in a hypermap which allow to count the spanning hypertrees of a hypermap recursively, and create a link with the computation of the Tutte polynomial of a graph. Having a particular interest in hypermaps which are reciprocals of maps, we generalize the reduction map introduced by Franz and Earnshaw to enumerate meanders to a reduction map that allows the enumeration of the spanning hypertrees of such hypermaps.