19 research outputs found
Independent transversals in locally sparse graphs
Let G be a graph with maximum degree \Delta whose vertex set is partitioned
into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G)
containing exactly one vertex from each part V_i. If it is also an independent
set, then we call it an independent transversal. The local degree of G is the
maximum number of neighbors of a vertex v in a part V_i, taken over all choices
of V_i and v \not \in V_i. We prove that for every fixed \epsilon > 0, if all
part sizes |V_i| >= (1+\epsilon)\Delta and the local degree of G is o(\Delta),
then G has an independent transversal for sufficiently large \Delta. This
extends several previous results and settles (in a stronger form) a conjecture
of Aharoni and Holzman. We then generalize this result to transversals that
induce no cliques of size s. (Note that independent transversals correspond to
s=2.) In that context, we prove that parts of size |V_i| >=
(1+\epsilon)[\Delta/(s-1)] and local degree o(\Delta) guarantee the existence
of such a transversal, and we provide a construction that shows this is
asymptotically tight.Comment: 16 page
The density Turan problem for 3-uniform linear hypertrees. An efficient testing algorithm
Let be a 3-uniform linear hypertree. We consider a blow-up hypergraph . We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph of the hypertree , with hyperedge densities satisfying some conditions, such that the hypertree does not appear in a blow-up hypergraph as a transversal. We present an efficient algorithm to decide whether a given set of hyperedge densities ensures the existence of a 3-uniform linear hypertree in a blow-up hypergraph
On the strong chromatic number of random graphs
Let G be a graph with n vertices, and let k be an integer dividing n. G is
said to be strongly k-colorable if for every partition of V(G) into disjoint
sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex
k-coloring of G with each color appearing exactly once in each V_i. In the case
when k does not divide n, G is defined to be strongly k-colorable if the graph
obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly
k-colorable. The strong chromatic number of G is the minimum k for which G is
strongly k-colorable. In this paper, we study the behavior of this parameter
for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove
that the strong chromatic number is a.s. concentrated on one value \Delta+1,
where \Delta is the maximum degree of the graph. We also obtain several weaker
results for sparse random graphs.Comment: 16 page
Complete subgraphs in a multipartite graph
In 1975 Bollob\'as, Erd\H os, and Szemer\'edi asked the following question:
given positive integers with , what is the largest
minimum degree among all -partite graphs with parts of size
and which do not contain a copy of ? The case has
attracted a lot of attention and was fully resolved by Haxell and Szab\'{o},
and Szab\'{o} and Tardos in 2006. In this paper we investigate the case
of the problem, which has remained dormant for over forty years. We resolve the
problem exactly in the case when , and up to an additive
constant for many other cases, including when . Our
approach utilizes a connection to the related problem of determining the
maximum of the minimum degrees among the family of balanced -partite
-vertex graphs of chromatic number at most
Extremal Problems For Transversals In Graphs With Bounded Degree
We introduce and discuss generalizations of the problem of independent transversals. Given a graph property {\user1{\mathcal{R}}} , we investigate whether any graph of maximum degree at most d with a vertex partition into classes of size at least p admits a transversal having property {\user1{\mathcal{R}}} . In this paper we study this problem for the following properties {\user1{\mathcal{R}}} : "acyclicâ, "H-freeâ, and "having connected components of order at most râ. We strengthen a result of [13]. We prove that if the vertex set of a d-regular graph is partitioned into classes of size d+âd/râ, then it is possible to select a transversal inducing vertex disjoint trees on at most r vertices. Our approach applies appropriate triangulations of the simplex and Sperner's Lemma. We also establish some limitations on the power of this topological method. We give constructions of vertex-partitioned graphs admitting no independent transversals that partially settles an old question of BollobĂĄs, ErdĆs and SzemerĂ©di. An extension of this construction provides vertex-partitioned graphs with small degree such that every transversal contains a fixed graph H as a subgraph. Finally, we pose several open question
THE DENSITY TURĂN PROBLEM
Let H be a graph on n vertices and let the blow-up graph G[H] be defined as follows. We replace each vertex vi of H by a cluster Ai and connect some pairs of vertices of Ai and Aj if (vi,vj) is an edge of the graph H. As usual, we define the edge density between Ai and Aj as d(Ai,Aj)= e(Ai,Aj)/|Ai||Aj|. We study the following problem. Given densities Îłij for each edge (i,j) âE(H), one has to decide whether there exists a blow-up graph G[H], with edge densities at least Îłij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic to H, i.e., no H appears as a transversal in G[H]. We call dcrit(H) the maximal value for which there exists a blow-up graph G[H] with edge densities d(Ai,Aj)=d crit(H) ((vi,vj) âE(H)) not containing H in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs. First, in the case of tree T we give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversal T in the blow-up graph. Then we give general bounds on dcrit(H) 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 H-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools. © 2012 Copyright Cambridge University Press