19 research outputs found
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
A graph is -free if it does not contain pairwise vertex-disjoint and
non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in
-free graphs can be solved in quasi-polynomial time. As a main technical
result, we establish that "sparse" (here, not containing large complete
bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback
vertex set number) at most logarithmic in the number of vertices. This is
proven sharp as there is an infinite family of -free graphs without
-subgraph and whose treewidth is (at least) logarithmic.
Other consequences include that most of the central NP-complete problems
(such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set,
Minimum Coloring) can be solved in polynomial time in sparse -free graphs,
and that deciding the -freeness of sparse graphs is polynomial time
solvable.Comment: 28 pages, 6 figures. v3: improved complexity result
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails
QPTAS and subexponential algorithm for maximum clique on disk graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time for \textsc{Maximum Clique} on disk graphs. In stark contrast, \textsc{Maximum Clique} on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time , unless the Exponential Time Hypothesis fails
Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture
In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any
-vertex graph can be decomposed into cycles and edges. We improve
upon the previous best bound of cycles and edges due to
Conlon, Fox and Sudakov, by showing an -vertex graph can always be
decomposed into cycles and edges, where is the
iterated logarithm function.Comment: Final version, accepted for publicatio
Towards the Erdős-Gallai cycle decomposition conjecture
In the 1960's, Erdős and Gallai conjectured that the edges of any n-vertex graph can be decomposed into O(n) cycles and edges. We improve upon the previous best bound of O(nloglogn) cycles and edges due to Conlon, Fox and Sudakov, by showing an n-vertex graph can always be decomposed into O(nlog∗n) cycles and edges, where log∗n is the iterated logarithm function
Packing and embedding large subgraphs
This thesis contains several embedding results for graphs in both random and non random settings.
Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals . %posed e.g.~by Bollob\'as,
In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n:
if H\subseteq\cQ^n satisfies with fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with fixed, then with high probability H\cup\cQ^n_p contains edge-disjoint Hamilton cycles, for any fixed .
This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity.
In Chapter 3 we move to a non random setting. %to a deterministic one.
%Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph.
Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs.
More specifically, we provide a degree condition on a regular -vertex graph which ensures the existence of a near optimal packing of any family of bounded degree -vertex -chromatic separable graphs into .
%In general, this degree condition is best possible.
%In particular, this yields an approximate version of the tree packing conjecture
%in the setting of regular host graphs of high degree.
%Similarly, our result implies approximate versions of the Oberwolfach problem,
%the Alspach problem and the existence of resolvable designs in the setting of
%regular host graphs of high degree.
In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem,
the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree