Isometric embeddings of Johnson graphs in Grassmann graphs

Let $V$ be an $n$-dimensional vector space ($4\le n <\infty$) and let ${\mathcal G}_{k}(V)$ be the Grassmannian formed by all $k$-dimensional subspaces of $V$. The corresponding Grassmann graph will be denoted by $\Gamma_{k}(V)$. We describe all isometric embeddings of Johnson graphs $J(l,m)$, $1<m<l-1$ in $\Gamma_{k}(V)$, $1<k<n-1$ (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of $J(n,k)$ in $\Gamma_{k}(V)$ is an apartment of ${\mathcal G}_{k}(V)$ if and only if $n=2k$. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in $\Gamma_{k}(V)$, $1<k<n-1$.Comment: New version -- 14 pages accepted to Journal of Algebraic Combinatoric

On Polygons Excluding Point Sets

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general position, and the convex hull of $B$ contains $k$ interior blue points and $l$ interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points $R$. We show that there is a minimal number $K=K(l)$, which is polynomial in $l$, such that one can always find a blue polygonization excluding all red points, whenever $k\geq K$. Some other related problems are also considered.Comment: 14 pages, 15 figure

Embeddings of Grassmann graphs

Let $V$ and $V'$ be vector spaces of dimension $n$ and $n'$, respectively. Let $k\in\{2,...,n-2\}$ and $k'\in\{2,...,n'-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph $\Gamma_{k}(V)$ in the Grassmann graph $\Gamma_{k'}(V')$

Product Dimension of Forests and Bounded Treewidth Graphs

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n vertices with a treewidth at most t has a product dimension at most (t+2)(logn+1). We also show that every k-degenerate graph on n vertices has a product dimension at most \ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by Eaton and Rodl.Comment: 12 pages, 3 figure