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8 Abstract A special case of a combinatorial theorem of De Bruijn and Erd˝ os asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by… (More)

For a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exists (a, b)-codes in Sierpi´nski graphs.

One of the De Bruijn-Erd˝ os theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to… (More)

It is already known that for very small edge cuts in graphs, the crossing number of the graph is at least the sum of the crossing number of (slightly augmented) components resulting from the cut. Under stronger connectivity condition in each cut component that was formalized as a graph operation called zip product, a similar result was obtained for edge… (More)

Given a directed graph D = (V, A) we define its intersection graph I(D) = (A, E) to be the graph having A as a node-set and two nodes of I(D) are adjacent if their corresponding arcs share a common node that is the tail of at least one of these arcs. We call these graphs facility location graphs since they arise from the classical uncapacitated facility… (More)

A collection of sets on a ground set Un (Un denotes the set {1, 2, ..., n}) closed under intersection and containing Un is known as a Moore family. The set of Moore families for a xed n is in bijection with the set of Moore co-families (union-closed families containing the empty set) denoted Mn. In this paper, we show that the set Mn can be endowed with the… (More)

- Laurent Beaudou
- 2009

We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K 4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K 4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K 4-minor free graph… (More)