DeepAI AI Chat
Log In Sign Up

On density of subgraphs of Cartesian products

by   Victor Chepoi, et al.

In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs G of Cartesian products G_1×...× G_m of arbitrary connected graphs. Namely, we show that |E(G)|/|V(G)|< 2{dens(G_1),...,dens(G_m)}|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) and VC-density VC-dens(G) of a subgraph G of a Cartesian product G_1×...× G_m, generalizing the classical Vapnik-Chervonenkis dimension of set-families (viewed as subgraphs of hypercubes). We prove that if G_1,...,G_m belong to the class G(H) of all finite connected graphs not containing a given graph H as a minor, then for any subgraph G of G_1×...× G_m a sharper inequality |E(G)|/|V(G)|<VC-dim(G)α(H) holds, where α(H) is the density of the graphs from G(H). We refine and sharpen those two results to several specific graph classes. We also derive upper bounds (some of them polylogarithmic) for the size of adjacency labeling schemes of subgraphs of Cartesian products.


page 1

page 2

page 3

page 4


Optimal Adjacency Labels for Subgraphs of Cartesian Products

For any hereditary graph class ℱ, we construct optimal adjacency labelin...

On density of subgraphs of halved cubes

Let S be a family of subsets of a set X of cardinality m and VC-dim( S)...

Excluding a ladder

Which graph classes C exclude a fixed ladder as a minor? We show that th...

Densest Subgraphs of a Dense Erdös-Rényi Graph. Asymptotics, Landscape and Universality

We consider the problem of estimating the edge density of densest K-node...

Construction of k-matchings and k-regular subgraphs in graph products

A k-matching M of a graph G=(V,E) is a subset M⊆ E such that each connec...

Efficient Minimax Signal Detection on Graphs

Several problems such as network intrusion, community detection, and dis...

Connections between graphs and matrix spaces

Given a bipartite graph G, the graphical matrix space 𝒮_G consists of ma...