ﻻ يوجد ملخص باللغة العربية
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 Mobius 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 Whitneys 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.
The abstract induced subgraph poset of a graph is the isomorphism class of the induced subgraph poset of the graph, suitably weighted by subgraph counting numbers. The abstract bond lattice and the abstract edge-subgraph poset are defined similarly b
Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform
Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The maximum n
The maximum size, $La(n,P)$, of a family of subsets of $[n]={1,2,...,n}$ without containing a copy of $P$ as a subposet, has been intensively studied. Let $P$ be a graded poset. We say that a family $mathcal{F}$ of subsets of $[n]={1,2,...,n}$ cont
In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contains any isolated arcs and any arc has length at least four. Modular diagrams represent the deformat