Related papers: Boxicity and separation dimension

Let $\mbox{$\cal F$}\subseteq 2^{[n]}$ be a fixed family of subsets. Let $D(\mbox{$\cal F$})$ stand for the following set of Hamming distances: $$ D(\mbox{$\cal F$}):=\{d_H(F,G):~ F, G\in \mbox{$\cal F$},\ F\neq G\}. $$ $\mbox{$\cal F$}$ is…

A Hamilton decomposition of a graph is a partitioning of its edge set into disjoint spanning cycles. The existence of such decompositions is known for all hypercubes of even dimension $2n$. We give a decomposition for the case $n = 2^a3^b$…

Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be…

This paper contains a comparative study of two families of simple curves drawn in the plane. On the one hand, we have the fractal curves on the unit interval, with self-similar structure, which have associated a Hausdorff dimension. On the…

The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by…

Suppose $k\nmid n$ and $H$ is an $n$-vertex $k$-uniform hypergraph. A near perfect matching in $H$ is a matching of size $\lfloor n/k\rfloor$. We give a divisibility barrier construction that prevents the existence of near perfect matchings…

For a finite set $X$, we say that a set $H\subseteq X$ crosses a partition ${\cal P}=(X_1,\dots,X_k)$ of $X$ if $H$ intersects $\min (|H|,k)$ partition classes. If $|H|\geq k$, this means that $H$ meets all classes $X_i$, whilst for…

The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We…

Let $box(G)$ be the boxicity of a graph $G$, $G[H_1,H_2,\ldots, H_n]$ be the $G$-generalized join graph of $n$-pairwise disjoint graphs $H_1,H_2,\ldots, H_n$, $G^d_k$ be a circular clique graph (where $k\geq 2d$) and $\Gamma(R)$ be the…

A set W \subseteq V (G) is called a resolving set, if for each pair of distinct vertices u,v \in V (G) there exists t \in W such that d(u,t) \neq d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum…

Let $k$ be a positive integer and let $D$ be a digraph. A path partition $\sP$ of $D$ is a set of vertex-disjoint paths which covers $V(D)$. Its $k$-norm is defined as $\sum_{P \in \sP} \Min{|V(P)|, k}$. A path partition is $k$-optimal if…

Let $G$ and $H$ be graphs, with $|V(H)|\geq |V(G)| $, and $f:V(G)\rightarrow V(H)$ a one to one map of their vertices. Let $dilation(f) = max\{ dist_{H}(f(x),f(y)): xy\in E(G) \}$, where $dist_{H}(v,w)$ is the distance between vertices $v$…

Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains $P$. Packings of maximum…

The boxicity of a graph $G$ is the minimum dimension $d$ that admits a representation of $G$ as the intersection graph of a family of axis-parallel boxes in $\mathbb{R}^d$. Computing boxicity is an NP-hard problem, and there are few known…

The aim of this paper is to extend and generalise some work of Katona on the existence of perfect matchings or Hamilton cycles in graphs subject to certain constraints. The most general form of these constraints is that we are given a…

A {\it mixed hypergraph} ${\cal H}=({\cal V},{\cal C},{\cal D})$ consists of the vertex set ${\cal V}$ and two families of subsets of $2^{{\cal V}}$: the family ${\cal C}$ of co-edges and the family ${\cal D}$ of edges. ${\cal H}$ is said…

Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the…

Let $A$ and $B$ be finite sets and consider a partition of the \emph{discrete box} $A \times B$ into \emph{sub-boxes} of the form $A' \times B'$ where $A' \subset A$ and $B' \subset B$. We say that such a partition has the…

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of…

Let $H$ be a $k$-uniform hypergraph on $n$ vertices where $n$ is a sufficiently large integer not divisible by $k$. We prove that if the minimum $(k-1)$-degree of $H$ is at least $\lfloor n/k \rfloor$, then $H$ contains a matching with…