Matching Complexes of Trees and Applications of the Matching Tree Algorithm

• Published in: Annals of combinatorics Vol. 26; no. 4; pp. 1041 - 1075
• Main Authors: Jelić Milutinović, Marija, Jenne, Helen, McDonough, Alex, Vega, Julianne
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-12-2022; Springer Nature B.V; Springer
• Subjects: Algorithms; caterpillar graph; Combinatorics; Graph theory; Graphs; homotopy type; honeycomb graph; Matching; matching complex; matching tree algorithm; Mathematics; MATHEMATICS AND COMPUTING; Mathematics and Statistics; Topology; Trees (mathematics); Homotopy type; Matching complex; Honeycomb graph; 05C70; Matching tree algorithm; 05E45; 05C69; Caterpillar graph
• ISSN: 0218-0006; 0219-3094
• DOI: 10.1007/s00026-022-00605-3

A matching complex of a simple graph
G
is a simplicial complex with faces given by the matchings of
G
. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa showed that matching complexes of forests are contractible or homotopy equivalent to a wedge of spheres. We study two specific families of trees. For caterpillar graphs, we give explicit formulas for the number of spheres in each dimension and for perfect binary trees we find a strict connectivity bound. We also use a tool from discrete Morse theory called the
Matching Tree Algorithm
to study the connectivity of honeycomb graphs, partially answering a question raised by Jonsson.