The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...

The Vapnik-Červonenkis dimension is a complexity measure of set-systems, or hypergraphs. Its application to graphs is usually done by considering the sets of neighborhoods of the vertices (see [1] and [5]), hence providing a set-system. But the graph structure is lost in the process. The aim of this paper is to introduce the notion of paired VC-dimension, a generalization of VC-dimension to set...

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadw...

For every d ≥ 3 and k ∈ {2} ∪ [3,∞), we determine the smallest ε such that every fractional (k + ε)-precoloring of vertices at mutual distance at least d of a graph G with fractional chromatic number equal to k can be extended to a proper fractional (k + ε)-coloring of G. Our work complements the analogous results of Albertson for ordinary colorings and those of Albertson and West for circular ...

In this note, we present a fractional version of Haemers' bound on the Shannon capacity of a graph, which is originally due to Blasiak. This bound is a common strengthening of both Haemers' bound and the fractional chromatic number of a graph. We show that this fractional version outperforms any bound on the Shannon capacity that could be attained through Haemers' bound. We show also that this ...

for a given hypergraph $h$ with chromatic number $chi(h)$ and with no edge containing only one vertex, it is shown that the minimum number $l$ for which there exists a partition (also a covering) ${e_1,e_2,ldots,e_l}$ for $e(h)$, such that the hypergraph induced by $e_i$ for each $1leq ileq l$ is $k$-colorable, is $lceil log_{k} chi(h) rceil$.

The rectangle covering number of an n × n Boolean matrix M is the smallest number of 1-rectangles which are needed to cover all the 1-entries of M . Its binary logarithm is the Nondeterministic Communication Complexity, and it equals the chromatic number of a graph G (M) obtained from M by a construction of Lovász and Saks. We determine the rectangle covering number and related parameters (cliq...