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Register a new userA construction of the abstract induced subgraph poset of a graph from its abstract edge subgraph poset
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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 by considering the lattice of subgraphs induced by connected partitions and the poset of edge-subgraphs, respectively. Continuing our development of graph reconstruction theory on these structures, we show that if a graph has no isolated vertices, then its abstract bond lattice and the abstract induced subgraph poset can be constructed from the abstract edge-subgraph poset except for the families of graphs that we characterise. The construction of the abstract induced subgraph poset from the abstract edge-subgraph poset generalises a well known result in reconstruction theory that states that the vertex deck of a graph with at least 4 edges and without isolated vertices can be constructed from its edge deck.12
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A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$.
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.
We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct unlabelled edge-subgraphs. We study these posets given up to isomorphism, and their relation to the reconstruction conjectures. We show that when G is not a star or a disjoint union of edges, P(G) and Omega(G) can be constructed from each other. The result implies that trees are reconstructible from their abstract bond lattice. We present many results on the reconstruction questions about the chromatic symmetric function and the symmetric Tutte polynomial. In particular, we show that the symmetric Tutte polynomial of a tree can be constructed from its chromatic symmetric function. We classify graphs that are not reconstructible from their abstract edge-subgraph posets, and further show that the families presented here are the only graphs not Q-reconstructible if and only if the edge reconstruction conjecture is true. Let f be a bijection from the set of all unlabelled graphs to itself such that for all unlabelled graphs G and H, hom(G,H) = hom(f(G), f(H)). We conjecture that f is an identity map. We show that this conjecture is weaker than the edge reconstruction conjecture. Our conjecture is motivated by homomorphism cancellation results due to Lovasz.
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 $[n]={1,2,...,n}$ contains a emph{rank-preserving} copy of $P$ if it contains a copy of $P$ such that elements of $P$ having the same rank are mapped to sets of same size in $mathcal{F}$. The largest size of a family of subsets of $[n]={1,2,...,n}$ without containing a rank-preserving copy of $P$ as a subposet is denoted by $La_{rp}(n,P)$. Clearly, $La(n,P) le La_{rp}(n,P)$ holds. In this paper we prove asymptotically optimal upper bounds on $La_{rp}(n,P)$ for tree posets of height $2$ and monotone tree posets of height $3$, strengthening a result of Bukh in these cases. We also obtain the exact value of $La_{rp}(n,{Y_{h,s},Y_{h,s}})$ and $La(n,{Y_{h,s},Y_{h,s}})$, where $Y_{h,s}$ denotes the poset on $h+s$ elements $x_1,dots,x_h,y_1,dots,y_s$ with $x_1<dots<x_h<y_1,dots,y_s$ and $Y_{h,s}$ denotes the dual poset of $Y_{h,s}$.
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 hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990s.
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