The Density Turán Problem

Abstract

<jats:p>Let<jats:italic>H</jats:italic>be a graph on<jats:italic>n</jats:italic>vertices and let the blow-up graph<jats:italic>G</jats:italic>[<jats:italic>H</jats:italic>] be defined as follows. We replace each vertex<jats:italic>v</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>of<jats:italic>H</jats:italic>by a cluster<jats:italic>A</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>and connect some pairs of vertices of<jats:italic>A<jats:sub>i</jats:sub></jats:italic>and<jats:italic>A<jats:sub>j</jats:sub></jats:italic>if (<jats:italic>v<jats:sub>i</jats:sub></jats:italic>,<jats:italic>v<jats:sub>j</jats:sub></jats:italic>) is an edge of the graph<jats:italic>H</jats:italic>. As usual, we define the edge density between<jats:italic>A<jats:sub>i</jats:sub></jats:italic>and<jats:italic>A<jats:sub>j</jats:sub></jats:italic>as<jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0963548312000016_eqnU1" /></jats:disp-formula>We study the following problem. Given densities γ<jats:sub><jats:italic>ij</jats:italic></jats:sub>for each edge (<jats:italic>i</jats:italic>,<jats:italic>j</jats:italic>) ∈<jats:italic>E</jats:italic>(<jats:italic>H</jats:italic>), one has to decide whether there exists a blow-up graph<jats:italic>G</jats:italic>[<jats:italic>H</jats:italic>], with edge densities at least γ<jats:sub><jats:italic>ij</jats:italic></jats:sub>, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic to<jats:italic>H</jats:italic>,<jats:italic>i.e.</jats:italic>, no<jats:italic>H</jats:italic>appears as a transversal in<jats:italic>G</jats:italic>[<jats:italic>H</jats:italic>]. We call<jats:italic>d</jats:italic><jats:sub>crit</jats:sub>(<jats:italic>H</jats:italic>) the maximal value for which there exists a blow-up graph<jats:italic>G</jats:italic>[<jats:italic>H</jats:italic>] with edge densities<jats:italic>d</jats:italic>(<jats:italic>A<jats:sub>i</jats:sub></jats:italic>,<jats:italic>A<jats:sub>j</jats:sub></jats:italic>)=<jats:italic>d</jats:italic><jats:sub>crit</jats:sub>(<jats:italic>H</jats:italic>) ((<jats:italic>v<jats:sub>i</jats:sub></jats:italic>,<jats:italic>v<jats:sub>j</jats:sub></jats:italic>) ∈<jats:italic>E</jats:italic>(<jats:italic>H</jats:italic>)) not containing<jats:italic>H</jats:italic>in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.</jats:p><jats:p>First, in the case of tree<jats:italic>T</jats:italic>we give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversal<jats:italic>T</jats:italic>in the blow-up graph. Then we give general bounds on<jats:italic>d</jats:italic><jats:sub>crit</jats:sub>(<jats:italic>H</jats:italic>) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction for<jats:italic>H</jats:italic>-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools.</jats:p>

The density Turán problem for some 3-uniform unihypercyclic linear hypergraphs. An efficient testing algorithm

“…We extend some results presented in [8], where authors studied the inhomogeneous problem for trees and proved the following theorem. Theorem 1.…”

“…This two problems has been studied in [8], [12] for simple graphs which are 2-uniform linear hypergraphs. We extend some of those results to 3-uniform linear hypergraphs with the hypercycle C 3 .…”

“…Theorem 1 provides authors of [8] with an efficient algorithm (Algorithm T ) to decide whether a given set of edge densities in a tree ensures the existence of a transversal or does not ensure. Their algorithm is cited on the next page.…”

“…Authors of [4] obtained Theorem 3 cited for completeness of this paper. Ealier this subject has been studied by Csikvári and Nagy [8] for trees and in [3] for some family of connected unicyclic graphs. Fig.…”

“…Ealier Csikvári and Nagy [8] discovered an interesting algorithm for testing whether a tree with a given set of edge densities is a factor of a blow-up graph. Some generalization of their algorithm is presented in [4].…”