Simplicial Embeddings between Pants Graphs

Vertex Decomposable Simplicial Complexes Associated to Path Graphs

Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...

On Isomorphism of Simplicial Complexes and Their Related Algebras

Labeling Subgraph Embeddings and Cordiality of Graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Geometric complexity of embeddings in ${\mathbb R}^d$

Given a simplicial complex K , we consider several notions of geometric complexity of embeddings of K in a Euclidean space R : thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any n-complex with N simplices which topologically embeds in R , n > 2, can be PL embedded in R with refinement complexity O(e 4+ǫ ). Families of s...

The Automorphism Group of the Complex of Pants Decompositions

There has been recent interest in finding combinatorial models for groups, i.e. finding a simplicial complex whose automorphism group is a given group. For example, CharneyDavis showed that the Coxeter Diagram is a model for Out(W ) (for certain Coxeter groups W) [3], and Bridson-Vogtmann showed that a spine of Outer Space is a model for Out(Fn) [1]. In this paper, we prove that the complex of ...