The Noncrossing Bond Poset of a Graph

TL;DR

This paper introduces the noncrossing bond poset for graphs, exploring its properties, conditions for lattice structure, and combinatorial invariants, extending classical combinatorial concepts to a noncrossing setting.

Contribution

It defines the noncrossing bond poset, analyzes its lattice properties, and provides combinatorial formulas for its invariants, extending known lattice theory to noncrossing partitions of graphs.

Findings

Noncrossing bond poset is a lattice for certain graph families.

Provides combinatorial formulas for Möbius function and characteristic polynomial.

Establishes conditions for shellability and supersolvability.

Abstract

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the M\"obius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.