173 research outputs found
Decompositions into subgraphs of small diameter
We investigate decompositions of a graph into a small number of low diameter
subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E)
on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that
|E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the
subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma,
Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a
constant depending only \epsilon. This shows that every dense graph can be
partitioned into a small number of ``small worlds'' provided that few edges can
be ignored. Improving on their result, we determine P(n,\epsilon,d) within an
absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded
for \epsilon
n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We
also prove that if G has large minimum degree, all the edges of G can be
covered by a small number of low diameter subgraphs. Finally, we extend some of
these results to hypergraphs, improving earlier work of Polcyn, R\"odl,
Ruci\'nski, and Szemer\'edi.Comment: 18 page
Improved Bounds for the Graham-Pollak Problem for Hypergraphs
For a fixed , let denote the minimum number of complete
-partite -graphs needed to partition the complete -graph on
vertices. The Graham-Pollak theorem asserts that . An easy
construction shows that ,
and we write for the least number such that .
It was known that for each even , but this was not known
for any odd value of . In this short note, we prove that . Our
method also shows that , answering another open problem
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Three notions of tropical rank for symmetric matrices
We introduce and study three different notions of tropical rank for symmetric
and dissimilarity matrices in terms of minimal decompositions into rank 1
symmetric matrices, star tree matrices, and tree matrices. Our results provide
a close study of the tropical secant sets of certain nice tropical varieties,
including the tropical Grassmannian. In particular, we determine the dimension
of each secant set, the convex hull of the variety, and in most cases, the
smallest secant set which is equal to the convex hull.Comment: 23 pages, 3 figure
Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs
For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ? V(G) such that G? S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) ? n^{?(1)} assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively.
We show that f_H(t) = 2^{?(t^{h-2})} for every graph H on h ? 3 vertices, and that f_H(t) = 2^{?(t)} if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^{?(t^{h-2})}. We also show that f_H(t) = 2^{?(t^{h})} when H = K_{h,h}, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_{C?}(t) for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h ? 2 the problem can be solved in polynomial time; if h ? 3, f_H(t) = 2^{?(t)} if H is a clique, and f_H(t) = 2^{?(t^{h-2})} otherwise
Hitting forbidden induced subgraphs on bounded treewidth graphs
For a fixed graph , the -IS-Deletion problem asks, given a graph ,
for the minimum size of a set such that does
not contain as an induced subgraph. Motivated by previous work about
hitting (topological) minors and subgraphs on bounded treewidth graphs, we are
interested in determining, for a fixed graph , the smallest function
such that -IS-Deletion can be solved in time assuming the Exponential Time Hypothesis (ETH), where and
denote the treewidth and the number of vertices of the input graph,
respectively.
We show that for every graph on
vertices, and that if is a clique or an independent
set. We present a number of lower bounds by generalizing a reduction of Cygan
et al. [MFCS 2014] for the subgraph version. In particular, we show that when
deviates slightly from a clique, the function suffers a sharp
jump: if is obtained from a clique of size by removing one edge, then
. We also show that
when , and this reduction answers an open question of Mi. Pilipczuk
[MFCS 2011] about the function for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant
of the problem, where each vertex of is colored with some color from
and we require to hit only induced copies of with matching colors. In this
case, we determine, under the ETH, the function for every connected
graph on vertices: if the problem can be solved in polynomial
time; if , if is a clique, and otherwise.Comment: 24 pages, 3 figure
Counting Connected Partitions of Graphs
Motivated by the theorem of Gy\H ori and Lov\'asz, we consider the following
problem. For a connected graph on vertices and edges determine the
number of unordered solutions of positive integers such that every is realized by a connected subgraph of with
edges such that . We also consider the
vertex-partition analogue.
We prove various lower bounds on as a function of the number of
vertices in , as a function of the average degree of , and also as
the size of -partite connected maximum cuts of .
Those three lower bounds are tight up to a multiplicative constant.
We also prove that the number of unordered -tuples with
, that are realizable by vertex partitions into
connected parts of respective sizes , is
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
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