The combinatorial dimension relative to an arbitrary fractional Cartesian product is defined. Relations between dimensions in certain archetypal instances are derived. Random sets with arbitrarily prescribed dimensions are produced; in particular, scales of combinatorial dimension are shown to be continuously and independently calibrated. A combinatorial concept of cylindricity is key. © 2004 W...

By Ulam’s conjecture every finite graph G can be reconstructed from its deck of vertex deleted subgraphs. The conjecture is still open, but many special cases have been settled. In particular, one can reconstruct Cartesian products. We consider the case of k-vertex deleted subgraphs of Cartesian products, and prove that one can decide whether a graph H is a kvertex deleted subgraph of a Cartesi...

We present an algorithm that determines the prime factors of connected graphs with respect to the Cartesian product in linear time and space. This improves a result of Aurenhammeer, Hagauer, and Imrich [2], who compute the prime factors in O(m log n) time, where m denotes the number of vertices of G and n the number of edges. Our algorithm is conceptually simpler. It gains its efficiency by the...

In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs G of Cartesian products G1 × · · · × Gm of arbitrary connected graphs. Namely, we show that |E(G)| |V (G)| ≤ d2 max{dens(G1), . . . , dens(Gm)}e log |V (G)|, where dens(H) is the maximum ratio |E(H′)| |V (H′)| over all subgraphs H ′ of H. We introduce the notions of VC-dimension VC-dim(G) an...

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau...

Packing measures Pg(E) and Hewitt-Stromberg measures νg(E) and their relatives are investigated. It is shown, for instance, that for any metric spaces X, Y and any Hausdorff functions f , g ν(X) ·P(Y ) 6 P(X × Y ). The inequality for the corresponding dimensions is established and used for a solution of a problem of Hu and Taylor: If X ⊆ Rn, then inf{dimPX × Y − dimPY : Y ⊆ R} = lim inf Xn↗X di...

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [4] conjectured tha...

A kernel J of a digraph D is an independent set of vertices of D such that for every vertex w ∈ V (D)\J there exists an arc from w to a vertex in J . In this paper we have obtained results for the existence and nonexistence of kernels in Cartesian products of certain families of digraphs, and characterized T Cn, T Pn and Cm Cn which have kernels, where T is a tournament, and Pn and Cn are, resp...

Cartesian products of graphs and hypergraphs have been studied since the 1960s. For (un)directed hypergraphs, unique prime factor decomposition (PFD) results with respect to the Cartesian product are known. However, there is still a lack of algorithms, that compute the PFD of directed hypergraphs with respect to the Cartesian product. In this contribution, we focus on the algorithmic aspects fo...