138 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Product structure of graph classes with strongly sublinear separators
We investigate the product structure of hereditary graph classes admitting
strongly sublinear separators. We characterise such classes as subgraphs of the
strong product of a star and a complete graph of strongly sublinear size. In a
more precise result, we show that if any hereditary graph class
admits separators, then for any fixed
every -vertex graph in is a subgraph
of the strong product of a graph with bounded tree-depth and a complete
graph of size . This result holds with if
we allow to have tree-depth . Moreover, using extensions of
classical isoperimetric inequalties for grids graphs, we show the dependence on
in our results and the above bound are
both best possible. We prove that -vertex graphs of bounded treewidth are
subgraphs of the product of a graph with tree-depth and a complete graph of
size , which is best possible. Finally, we investigate the
conjecture that for any hereditary graph class that admits
separators, every -vertex graph in is a
subgraph of the strong product of a graph with bounded tree-width and a
complete graph of size . We prove this for various classes
of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial
Expansion Classes" which had an error, added section "Lower Bounds", and
added a new author; v4: minor revisions and corrections
HM 32: New Interpretations in Naval History
Selected papers from the twenty-first McMullen Naval History Symposium held at the U.S. Naval Academy, 19–20 September 2019.https://digital-commons.usnwc.edu/usnwc-historical-monographs/1031/thumbnail.jp
Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring
The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have n ‘bad events’, each of which occurs with probability at most p and is independent of all but d other events, then under certain criteria on p and d, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM ’10] also implied an efficient distributed algorithm for the problem, running in O(log2 n) rounds. For instances with low d, this was improved to O(d 2 + logO(1) log n) by Fischer and Ghaffari [DISC ’17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP ’19]). We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating p and d, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari’s algorithm, we improve the round complexity to O( d log d + logO(1) log n). At the other end of the trade-off, we obtain a logO(1) log n round complexity for a substantially wider regime than previously known. As our main application, we also give the first logO(1) log n-round distributed algorithm for the problem of ∆+o(∆)-edge coloring a graph of maximum degree ∆. This is an almost exponential improvement over previous results: no prior logo(1) n-round algorithm was known even for 2∆ − 2-edge coloring
On the choosability of -minor-free graphs
Given a graph , let us denote by and ,
respectively, the maximum chromatic number and the maximum list chromatic
number of -minor-free graphs. Hadwiger's famous coloring conjecture from
1943 states that for every . In contrast, for list
coloring it is known that
and thus, is bounded away from the conjectured value for
by at least a constant factor. The so-called -Hadwiger's
conjecture, proposed by Seymour, asks to prove that
for a given graph (which would be implied by Hadwiger's conjecture). In
this paper, we prove several new lower bounds on , thus exploring
the limits of a list coloring extension of -Hadwiger's conjecture. Our main
results are:
For every and all sufficiently large graphs we have
, where
denotes the vertex-connectivity of .
For every there exists such that
asymptotically almost every -vertex graph with edges satisfies .
The first result generalizes recent results on complete and complete
bipartite graphs and shows that the list chromatic number of -minor-free
graphs is separated from the natural lower bound by a
constant factor for all large graphs of linear connectivity. The second
result tells us that even when is a very sparse graph (with an average
degree just logarithmic in its order), can still be separated from
by a constant factor arbitrarily close to . Conceptually
these results indicate that the graphs for which is close to
are typically rather sparse.Comment: 14 page
On Two problems of defective choosability
Given positive integers , and a non-negative integer , we say a
graph is -choosable if for every list assignment with
for each and ,
there exists an -coloring of such that each monochromatic subgraph has
maximum degree at most . In particular, -choosable means
-colorable, -choosable means -choosable and
-choosable means -defective -choosable. This paper proves
that there are 1-defective 3-choosable graphs that are not 4-choosable, and for
any positive integers , and non-negative integer , there
are -choosable graphs that are not -choosable.
These results answer questions asked by Wang and Xu [SIAM J. Discrete Math. 27,
4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353],
respectively. Our construction of -choosable but not -choosable graphs generalizes the construction of Kr\'{a}l' and Sgall
in [J. Graph Theory 49, 3(2005), 177-186] for the case .Comment: 12 pages, 4 figure
The grid-minor theorem revisited
We prove that for every planar graph of treedepth , there exists a
positive integer such that for every -minor-free graph , there exists
a graph of treewidth at most such that is isomorphic to a
subgraph of . This is a qualitative strengthening of the
Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the
optimal parameter in such a result. As an example application, we use this
result to improve the upper bound for weak coloring numbers of graphs excluding
a fixed graph as a minor
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Defective coloring is perfect for minors
The defective chromatic number of a graph class is the infimum such that
there exists an integer such that every graph in this class can be
partitioned into at most induced subgraphs with maximum degree at most .
Finding the defective chromatic number is a fundamental graph partitioning
problem and received attention recently partially due to Hadwiger's conjecture
about coloring minor-closed families. In this paper, we prove that the
defective chromatic number of any minor-closed family equals the simple lower
bound obtained by the standard construction, confirming a conjecture of Ossona
de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable
finite minors for infinite graphs that cannot be partitioned into a fixed
finite number of induced subgraphs with uniformly bounded maximum degree. As
corollaries about clustered coloring, we obtain a linear relation between the
clustered chromatic number of any minor-closed family and the tree-depth of its
forbidden minors, improving an earlier exponential bound proved by Norin,
Scott, Seymour, and Wood and confirming the planar case of their conjecture
Efficient Classification of Locally Checkable Problems in Regular Trees
We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the [Θ(log n), Θ(n)] region, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees. The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in O(log n) rounds. If not, it is known that the complexity has to be Θ(n^{1/k}) for some k = 1, 2, ..., and in this case the algorithms also output the right value of the exponent k.
In rooted trees in the O(log n) case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the O(log n) region remains an open question
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