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Register a new userOn Connectivity of the Facet Graphs of Simplicial Complexes
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The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$, $d$, and $(n-d-1)$-vertex-connected. It is also shown that the facet graph of a $d$-cycle cannot be split into more than $s$ connected components by removing at most $s$ vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.
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In this paper we study a pair of numerical parameters associated to a graph $G$. One the one hand, one can construct $text{Hom}(K_2, G)$, a space of homomorphisms from a edge $K_2$ into $G$ and study its (topological) connectivity. This approach dates back to the neighborhood complexes introduced by Lovasz in his proof of the Kneser conjecture. In another direction Brightwell and Winkler introduced a graph parameter called the warmth $zeta(G)$ of a graph $G$, based on asymptotic behavior of $d$-branching walks in $G$ and inspired by constructions in statistical physics. Both the warmth of $G$ and the connectivity of $text{Hom}(K_2,G)$ provide lower bounds on the chromatic number of $G$. Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph $G$ is always less than three plus the connectivity of $text{Hom}(K_2, G)$. We succeed in establishing a first nontrivial case of the conjecture, by showing that $zeta(G) leq 3$ if $text{Hom}(K_2,G)$ has an infinite first homology group. We also calculate warmth for a family of `twisted toroidal graphs that are important extremal examples in the context of $text{Hom}$ complexes. Finally we show that $zeta(G) leq n-1$ if a graph $G$ does not have the complete bipartite graph $K_{a,b}$ for $a+b=n$. This provides an analogue for a similar result in the context of $text{Hom}$ complexes.
A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian (Adv. in Math., 218 (2008), 87-104) in the framework of posets and we give the relation between the two approaches (theorems 3.5 and 3.7). We conclude with a question about the relation between the s-homotopy and the graph homotopy defined by Chen, Yau and Yeh (Discrete Math., 241(2001), 153-170).
Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ fail independently with probability $q in[0,1]$. The emph{all-terminal reliability} of $G$ is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in $q$, and it was conjectured cite{BC1} that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few and far between, and the roots are only barely outside the disk. In this paper we generalize the notion of reliability to simplicial complexes and matroids and investigate when, for small simplicial complexes and matroids, the roots fall inside the closed unit disk.
Let $L_n$ be a line graph with $n$ edges and $F(L_n)$ the facet ideal of its matching complex. In this paper, we provide the irreducible decomposition of $F(L_n)$ and some exact formulas for the projective dimension and the regularity of $F(L_n)$.
We provide a simple characterization of simplicial complexes on few vertices that embed into the $d$-sphere. Namely, a simplicial complex on $d+3$ vertices embeds into the $d$-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen--Flores theorem and provide a topological extension of the ErdH os--Ko--Rado theorem. By analogy with Farys theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.
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