Let k, d, λ > 1 be integers with d > λ and let X ⊂ R be a finite set. A (d−λ)-plane L transversal to the convex hull of all k-sets of X is called Kneser transversal. If in addition L contains (d− λ) + 1 points of X, then L is called complete Kneser transversal. In this paper, we present various results on the existence of (complete) Kneser transversals for λ = 2, 3. In order to do this, we intr...

Associated with every graph G of chromatic number χ is another graph G. The vertex set of G consists of all χ-colorings of G, and two χ-colorings are adjacent when they differ on exactly one vertex. According to a conjecture of Björner and Lovász, this graph G must be disconnected. In this note we give a counterexample to this conjecture. One of the most disturbing problems in graph theory is t...

We give a short proof for Chen’s Alternative Kneser Coloring Lemma. This leads to a short proof for the Johnson-Holroyd-Stahl conjecture that Kneser graphs have their circular chromatic numbers equal to their chromatic numbers.

In this paper some coloring properties of graphs power have been presented. In this regard, the helical graphs have been introduced and it was shown that they are hom-universal with respect to high odd-girth graphs whose lth power is bounded with a Kneser graph. In the sequel, we have considered Pentagon’s problem of Nešetřil which this problem is about the existence of high girth cubic graphs ...

1. Tue 26 July, 9-9:30 Marcia Fampa, COPPE, Federal University of Rio de Janeiro, Brazil Modeling the Euclidean Steiner Tree problem In the Euclidean Steiner Tree Problem, the goal is to find a network of minimum length interconnecting a set P of given points in the n-dimensional Euclidean space. Such networks may be represented by a tree T , where the set of nodes is given by the points in P ,...

The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the total weight of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle...

A set A ⊆ {1, 2, . . . , n} is said to be k-separated if, when considered on the circle, any two elements of A are separated by a gap of size at least k. We prove a conjecture due to Holroyd and Johnson [3],[4] that an analogue of the Erdős-Ko-Rado theorem holds for k-separated sets. In particular the result holds for the vertex-critical subgraph of the Kneser graph identified by Schrijver [7],...

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the ...

We show that every core graph with a primitive automorphism group has the property that whenever it is a retract of a product of connected graphs, it is a retract of a factor. The example of Kneser graphs shows that the hypothesis that the factors are connected is essential. In the case of complete graphs, our result has already been shown in [4, 17], and it is an instance where Hedetniemi’s co...