A few words about maps

TL;DR

This paper explores combinatorial maps, their representations, and properties, introducing new bijections and decompositions that advance understanding of map enumeration and topological structures.

Contribution

It introduces novel bijections between combinatorial maps, chord diagrams, and double occurrence words, and develops a topological representation of maps with spanning quasi-trees.

Findings

Derived a topological representation of maps with quasi-trees.

Established bijections connecting maps, chord diagrams, and words.

Applied these results to enumerate loopless rooted maps.

Abstract

In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite circle graphs. Then, we focus on Depth-First Search trees and their connection with a poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the first section to the problem of enumerating loopless rooted maps. Finally, we return to the planar case and discuss a decomposition of planar rooted loopless maps and its consequences on planar rooted loopless map enumeration.