3,979 research outputs found
χ-bounded families of oriented graphs
A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets P of orientations of P 4 (the path on four vertices) similar statements hold. We establish some positive and negative results
Induced subgraphs of graphs with large chromatic number. XI. Orientations
Fix an oriented graph H, and let G be a graph with bounded clique number and
very large chromatic number. If we somehow orient its edges, must there be an
induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for
two specific kinds of digraph H: the three-edge path, with the first and last
edges both directed towards the interior; and stars (with many edges directed
out and many directed in). Aboulker et al subsequently conjectured that the
answer is affirmative in both cases. We give affirmative answers to both
questions
Testing bounded arboricity
In this paper we consider the problem of testing whether a graph has bounded
arboricity. The family of graphs with bounded arboricity includes, among
others, bounded-degree graphs, all minor-closed graph classes (e.g. planar
graphs, graphs with bounded treewidth) and randomly generated preferential
attachment graphs. Graphs with bounded arboricity have been studied extensively
in the past, in particular since for many problems they allow for much more
efficient algorithms and/or better approximation ratios.
We present a tolerant tester in the sparse-graphs model. The sparse-graphs
model allows access to degree queries and neighbor queries, and the distance is
defined with respect to the actual number of edges. More specifically, our
algorithm distinguishes between graphs that are -close to having
arboricity and graphs that -far from having
arboricity , where is an absolute small constant. The query
complexity and running time of the algorithm are
where denotes
the number of vertices and denotes the number of edges. In terms of the
dependence on and this bound is optimal up to poly-logarithmic factors
since queries are necessary (and .
We leave it as an open question whether the dependence on can be
improved from quasi-polynomial to polynomial. Our techniques include an
efficient local simulation for approximating the outcome of a global (almost)
forest-decomposition algorithm as well as a tailored procedure of edge
sampling
Clique versus Independent Set
Yannakakis' Clique versus Independent Set problem (CL-IS) in communication
complexity asks for the minimum number of cuts separating cliques from stable
sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial
CS-separator, i.e. of size , and addresses the problem of
finding a polynomial CS-separator. This question is still open even for perfect
graphs. We show that a polynomial CS-separator almost surely exists for random
graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a
clique and a stable set) then there exists a constant for which we find a
CS-separator on the class of H-free graphs. This generalizes a
result of Yannakakis on comparability graphs. We also provide a
CS-separator on the class of graphs without induced path of length k and its
complement. Observe that on one side, is of order
resulting from Vapnik-Chervonenkis dimension, and on the other side, is
exponential.
One of the main reason why Yannakakis' CL-IS problem is fascinating is that
it admits equivalent formulations. Our main result in this respect is to show
that a polynomial CS-separator is equivalent to the polynomial
Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition
into k complete bipartite graphs, then its chromatic number is polynomially
bounded in terms of k. We also show that the classical approach to the stubborn
problem (arising in CSP) which consists in covering the set of all solutions by
instances of 2-SAT is again equivalent to the existence of a
polynomial CS-separator
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