We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices forms a dominating set. We then construct three new infinite families of well-covered graphs, none of which have this property. We use these results to provide a minimal counterexample to a conjecture of Villarreal regarding Cohen-Macaulay graphs.
Let $G=(V,E)$ be a graph. If $G$ is a König graph or $G$ is a graph without 3-cycles and 5-cycle, we prove that the following conditions are equivalent: $Δ_{G}$ is pure shellable, $R/I_Δ$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of König type $e_{1},...,e_{g}$ without square with two $e_i$'s. We characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also, we study when graphs without 3-cycles and 5-cycles are vertex decomposable or shellable. Furthermore, we give some properties and relations between critical…
We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly triangle divisible graph that is also triangle decomposable. We explore several classes of planar graphs as well as a class of toroidal graphs.
We study the independence complexes of families of well-covered circulant graphs discovered by Boros-Gurvich-Milanič, Brown-Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g. vertex decomposable, shellable) or topological (e.g. Cohen-Macaulay, Buchsbaum) structure. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen-Macaulay, and/or Buchsbaum. …
In 1980 Provan and Billera defined the notion of weak $k$-decomposability for pure simplicial complexes. They showed the diameter of a weakly $k$-decomposable simplicial complex $Δ$ is bounded above by a polynomial function of the number of $k$-faces in $Δ$ and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of vertices and the dimension. In this paper we exhibit the first examples of non-weakly 0-decomposable simplicial polytopes.
Barnette's conjecture is an unsolved problem in graph theory. The problem states that every 3-regular (cubic), 3-connected, planar, bipartite (Barnette) graph is Hamiltonian. Partial results have been derived with restrictions on number of vertices, several properties of face-partitions and dual graphs of Barnette graphs while some studies focus just on structural characterizations of Barnette graphs. Noting that Spider web graphs are a subclass of Annular Decomposable Barnette (ADB graphs) graphs and are Hamiltonian, we study ADB graphs and their annular-connected subclass (ADB-AC graphs). We…
The junction-tree representation provides an attractive structural property for organizing a decomposable graph. In this study, we present two novel stochastic algorithms, which we call the junction-tree expander and junction-tree collapser for sequential sampling of junction trees for decomposable graphs. We show that recursive application of the junction-tree expander, expanding incrementally the underlying graph with one vertex at a time, has full support on the space of junction trees with any given number of underlying vertices. On the other hand, the junction-tree collapser provides a co…
We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$ satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty or ${\mathbf K}$ becomes collapsible after removing $\tilde β_d ({\mathbf K}; {\mathbb Z}_2)$ facets, where $\tilde β_d ({\mathbf K}; {\mathbb Z}_2)$ denotes the $d$th reduced Betti number. In this paper, we show that if the link of each face of a pure simplicial complex ${\mathbf K}$ (including the link of the empty face which is the whole ${\mathbf K}$) satisfy the removal-collapsibility condition, then the second barycentric subdivision of ${\m…
The 2-block intersection graph (2-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements. The 2-BIGs are themselves interesting graphs: each component is cubic and 3-connected, and a 2-BIG is bipartite exactly when the TTS is decomposable to two Steiner triple systems. Any connected bipartite 2-BIG with no Hamilton cycle is a counter-example to a conjecture posed by Tutte in 1971. Our main result is that there exists an integer $N$ such that for…
A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable…
A realisation of a metric $d$ on a finite set $X$ is a weighted graph $(G,w)$ whose vertex set contains $X$ such that the shortest-path distance between elements of $X$ considered as vertices in $G$ is equal to $d$. Such a realisation $(G,w)$ is called optimal if the sum of its edge weights is minimal over all such realisations. Optimal realisations always exist, although it is NP-hard to compute them in general, and they have applications in areas such as phylogenetics, electrical networks and internet tomography. In [Adv. in Math. 53, 1984, 321-402] A.~Dress showed that the optimal realisati…
In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $Δ_χ$ from a coloring $χ$ of a subset of the vertices of $Δ$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.
In a graph $G=(V,E)$, a module is a vertex subset $M$ of $V$ such that every vertex outside $M$ is adjacent to all or none of $M$. For example, $\emptyset$, $\{x\}$ $(x\in V )$ and $V$ are modules of $G$, called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex $x$ of a prime graph $G$ is critical if $G - x$ is decomposable. Moreover, a prime graph with $k$ non-critical vertices is called $(-k)$-critical graph. A prime graph $G$ is $k$-minimal if there is some $k$-vertex set $X$ of vertices such that there is no proper induced sub…
We describe two constructions of (very) dense graphs which are edge disjoint unions of large {\em induced} matchings. The first construction exhibits graphs on $N$ vertices with ${N \choose 2}-o(N^2)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o(1)}$. The second construction provides a covering of all edges of the complete graph $K_N$ by two graphs, each being the edge disjoint union of at most $N^{2-δ}$ induced matchings, where $δ> 0.058$. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linia…
Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number $χ_t(G)$ is the least integer $k$ for which $G$ admits a coloring with $k$ colors such that each color class induces a $(t-1)$-degenerate subgraph of $G$. So $χ_1$ is the chromatic number and $χ_2$ is the point aboricity. The point partition number $χ_t$ with $t\geq 1$ was introduced by Lick and White. A graph $G$ is called $χ_t$-critical if every proper subgraph $H$ of $G$ satisfies $χ_t(H)<χ_t(G)$. In this paper we prove that if $G$ is a $χ_t$-critical graph whose orde…
An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph on $N$ vertices has a monochromatic copy of $\mathcal{G}$ that respects the ordering. In this paper we investigate the effect of various graph operations on the Ramsey number of a given ordered graph, and detail a general framework for applying results on extremal functions of 0-1 matrices to ordered Ramsey problems. We apply this method to give upper bounds…
Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $ x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of $G$, called trivial intervals. A graph whose intervals are trivial is indecomposable; otherwise, it is decomposable. According to Ille, the indecomposability graph of an undirected indecomposable graph $G$ is the graph $\mathbb I(G)$ whose vertices are those of $G$ and edges are the unordered pairs of distinct vertices $\{x,y\}$ such that the induced subgra…
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ide…
We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opp…
Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set. A fundamental result of Wilson is that for all $n=|V(G)|$ sufficiently large, $G$ has a $k$-decomposition if and only if $G$ is $k$-divisible. Let ${\bf v} \in {\mathbb R}^{|F_k|}$ be indexed by $F_k$. For a $k$-decomposition $L$ of $G$, let $ν_{\bf v}(L) = \sum_{F \in F_k} {\bf v}_F d_{L,F}$ where $d_{L,F}$ is the fraction of elements of $L$ isomorphic to $…
A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples, 1975). A graph G belongs to class W_{n} if every n pairwise disjoint independent sets in G are included in $n$ pairwise disjoint maximum independent sets (J. W. Staples, 1975). Clearly, W_{1} is the family of all well-covered graphs. It turns out that G belongs to W_{2} if and only if it is a 1-well-covered graph without isolated vertices. We show that deleting …
Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on $n$ vertices is bounded by ${2^{\lceil 3.59n\rceil}}$. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration of distance-hereditary graphs, which allows to show that th…
Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number $π(G)$ is the smallest number such that, for any distribution of $π(G)$ pebbles to the vertices of $G$ and choice of root vertex $r$ of $G$, there exists a sequence of pebbling moves that places a pebble on $r$. Computing $π(G)$ is provably difficult, and recent methods for bounding $π(G)$ have proved computationally intractable, even for moderately sized graphs. Graham conjectured that $π(G ~\square~ H) \leq …
