Distance two labeling of direct product of paths and cycles

Suppose that $[n]=\left\{0,1,2,...,n\right\}$ is a set of non-negative integers and $h,k \in [n]$. The $L(h,k)$-labeling of graph $G$ is the function $l:V(G)\rightarrow[n]$ such that $\left|l(u)-l(v)\right|\geq h$ if the distance $d(u,v)$ between $u$ and $v$ is one and $\left|l(u)-l(v)\right| \geq k$ if the distance $d(u,v)$ is two. Let $L(V(G))=\left\{l(v): v \in V(G)\right\}$ and let $p$ be the maximum value of $L(V(G)).$ Then $p$ is called $\lambda_h^k-$number of $G$ if $p$ is the least possible member of $[n]$ such that $G$ maintains an $L(h,k)-$labeling. In this paper, we establish $\lambda_1^1-$ numbers of $P _m \times C_n$ graphs for all $m \geq 2$ and $n\geq 3$.Comment: 13 pages, 9 figure

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure

Cartesian products as profinite completions

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.Comment: latex 13 page

Extensions of a result of Elekes and R\'onyai

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an $n\times n\times n$ cartesian product in $\mathbb{R}^3$, then the polynomial has the form $f(x,y)=g(k(x)+l(y))$ or $f(x,y)=g(k(x)l(y))$. They used this to prove a conjecture of Purdy which states that given two lines in $\mathbb{R}^2$ and $n$ points on each line, if the number of distinct distances between pairs of points, one on each line, is at most $cn$, then the lines are parallel or orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on an $n\times n\times n\times n$ cartesian product and an asymmetric cartesian product. We give a proof of a variation of Purdy's conjecture with fewer points on one of the lines. We finish with a lower bound for our main result in one dimension higher with asymmetric cartesian product, showing that it is near-optimal.Comment: 23 page

Free quasi-symmetric functions, product actions and quantum field theory of partitions

We examine two associative products over the ring of symmetric functions related to the intransitive and Cartesian products of permutation groups. As an application, we give an enumeration of some Feynman type diagrams arising in Bender's QFT of partitions. We end by exploring possibilities to construct noncommutative analogues.Comment: Submitted 28.11.0