16 research outputs found
Polynomial treewidth forces a large grid-like-minor
Robertson and Seymour proved that every graph with sufficiently large
treewidth contains a large grid minor. However, the best known bound on the
treewidth that forces an grid minor is exponential in .
It is unknown whether polynomial treewidth suffices. We prove a result in this
direction. A \emph{grid-like-minor of order} in a graph is a set of
paths in whose intersection graph is bipartite and contains a
-minor. For example, the rows and columns of the
grid are a grid-like-minor of order . We prove that polynomial
treewidth forces a large grid-like-minor. In particular, every graph with
treewidth at least has a grid-like-minor of order
. As an application of this result, we prove that the cartesian product
contains a -minor whenever has treewidth at least
.Comment: v2: The bound in the main result has been improved by using the
Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte
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
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
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
Graphs of low average degree without independent transversals
An independent transversal of a graph G with a vertex partition P is an independent set of G intersecting each block of P in a single vertex. Wanless and Wood proved that if each block of P has size at least t and the average degree of vertices in each block is at most t/4, then an independent transversal of P exists. We present a construction showing that this result is optimal: for any Δ>0 and sufficiently large t, there is a family of forests with vertex partitions whose block size is at least t, average degree of vertices in each block is at most (1/4+Δ)t, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version
Approximate packing of independent transversals in locally sparse graphs
Consider a multipartite graph with maximum degree at most , parts
have size , and every vertex has at most
neighbors in any part . Loh and Sudakov proved that any such has an
independent transversal. They further conjectured that the vertex set of
can be decomposed into pairwise disjoint independent transversals. In the
present paper, we resolve this conjecture approximately by showing that
contains pairwise disjoint independent transversals. As applications,
we give approximate answers to questions of Yuster, and of Fischer, K\"uhn, and
Osthus