Counting Shellings of Complete Bipartite Graphs and Trees

TL;DR

This paper investigates shellings of graphs, providing exact formulas for complete bipartite graphs and bounds for trees, linking shelling numbers to poset linear extensions and graph parameters.

Contribution

It derives an exact formula for shelling numbers of complete bipartite graphs and establishes bounds for trees based on their structural properties.

Findings

Exact shelling number formula for complete bipartite graphs

Bounds on shelling numbers for trees using degrees and diameter

Connection between shellings of trees and linear extensions of posets

Abstract

A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite graphs and trees. For complete bipartite graphs, we obtain an exact formula for their shelling numbers. And for trees, we relate their shelling numbers to linear extensions of tree posets and bound shelling numbers using vertex degrees and diameter.