12,864 research outputs found
Local resilience of an almost spanning -cycle in random graphs
The famous P\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\'{o}s,
S\'{a}rk\"{o}zy, and Szemer\'{e}di, states that for any , every graph
on vertices with minimum degree contains the -th power of a
Hamilton cycle. We extend this result to a sparse random setting.
We show that for every there exists such that if then w.h.p. every subgraph of a random graph with
minimum degree at least , contains the -th power of a
cycle on at least vertices, improving upon the recent results of
Noever and Steger for , as well as Allen et al. for .
Our result is almost best possible in three ways: for the
random graph w.h.p. does not contain the -th power of any long
cycle; there exist subgraphs of with minimum degree and vertices not belonging to triangles; there exist
subgraphs of with minimum degree which do not
contain the -th power of a cycle on vertices.Comment: 24 pages; small updates to the paper after anonymous reviewers'
report
Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs
P\'osa's theorem states that any graph whose degree sequence satisfies for all has a Hamilton cycle.
This degree condition is best possible. We show that a similar result holds for
suitable subgraphs of random graphs, i.e. we prove a `resilience version'
of P\'osa's theorem: if and the -th vertex degree (ordered
increasingly) of is at least for all ,
then has a Hamilton cycle. This is essentially best possible and
strengthens a resilience version of Dirac's theorem obtained by Lee and
Sudakov.
Chv\'atal's theorem generalises P\'osa's theorem and characterises all degree
sequences which ensure the existence of a Hamilton cycle. We show that a
natural guess for a resilience version of Chv\'atal's theorem fails to be true.
We formulate a conjecture which would repair this guess, and show that the
corresponding degree conditions ensure the existence of a perfect matching in
any subgraph of which satisfies these conditions. This provides an
asymptotic characterisation of all degree sequences which resiliently guarantee
the existence of a perfect matching.Comment: To appear in the Electronic Journal of Combinatorics. This version
corrects a couple of typo
Embedding spanning bounded degree graphs in randomly perturbed graphs
We study the model G 8 G(n; p) of randomly perturbed dense graphs, where G is any n-vertex graph with minimum degree at least n and G(n; p) is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every > 0 and C 5, and every n-vertex graph F with maximum degree at most , we show that if p = !(n−2~(+1)) then G 8 G(n; p) with high probability contains a copy of F. The bound used for p here is lower by a log-factor in comparison to the conjectured threshold for the general appearance of such subgraphs in G(n; p) alone, a typical feature of previous results concerning randomly perturbed dense graphs. We also give the rst example of graphs where the appearance threshold in G 8 G(n; p) is lower than the appearance threshold in G(n; p) by substantially more than a log-factor. We prove that, for every k C 2 and > 0, there is some > 0 for which the kth power of a Hamilton cycle with high probability appears in G 8 G(n; p) when p = !(n−1~k−). The appearance threshold of the kth power of a Hamilton cycle in G(n; p) alone is known to be n−1~k, up to a log-term when k = 2, and exactly for k > 2
Robust Hamiltonicity in families of Dirac graphs
A graph is called Dirac if its minimum degree is at least half of the number
of vertices in it. Joos and Kim showed that every collection
of Dirac graphs on the same vertex set of
size contains a Hamilton cycle transversal, i.e., a Hamilton cycle on
with a bijection such that
for every .
In this paper, we determine up to a multiplicative constant, the threshold
for the existence of a Hamilton cycle transversal in a collection of random
subgraphs of Dirac graphs in various settings. Our proofs rely on constructing
a spread measure on the set of Hamilton cycle transversals of a family of Dirac
graphs.
As a corollary, we obtain that every collection of Dirac graphs on
vertices contains at least different Hamilton cycle transversals
for some absolute constant . This is optimal up to the constant
. Finally, we show that if is sufficiently large, then every such
collection spans pairwise edge-disjoint Hamilton cycle transversals, and
this is best possible. These statements generalize classical counting results
of Hamilton cycles in a single Dirac graph
Abelian networks IV. Dynamics of nonhalting networks
An abelian network is a collection of communicating automata whose state
transitions and message passing each satisfy a local commutativity condition.
This paper is a continuation of the abelian networks series of Bond and Levine
(2016), for which we extend the theory of abelian networks that halt on all
inputs to networks that can run forever. A nonhalting abelian network can be
realized as a discrete dynamical system in many different ways, depending on
the update order. We show that certain features of the dynamics, such as
minimal period length, have intrinsic definitions that do not require
specifying an update order.
We give an intrinsic definition of the \emph{torsion group} of a finite
irreducible (halting or nonhalting) abelian network, and show that it coincides
with the critical group of Bond and Levine (2016) if the network is halting. We
show that the torsion group acts freely on the set of invertible recurrent
components of the trajectory digraph, and identify when this action is
transitive.
This perspective leads to new results even in the classical case of sinkless
rotor networks (deterministic analogues of random walks). In Holroyd et. al
(2008) it was shown that the recurrent configurations of a sinkless rotor
network with just one chip are precisely the unicycles (spanning subgraphs with
a unique oriented cycle, with the chip on the cycle). We generalize this result
to abelian mobile agent networks with any number of chips. We give formulas for
generating series such as where is the number of recurrent chip-and-rotor configurations with
chips; is the diagonal matrix of outdegrees, and is the adjacency
matrix. A consequence is that the sequence completely
determines the spectrum of the simple random walk on the network.Comment: 95 pages, 21 figure
Long paths and cycles in random subgraphs of graphs with large minimum degree
For a given finite graph of minimum degree at least , let be a
random subgraph of obtained by taking each edge independently with
probability . We prove that (i) if for a function
that tends to infinity as does, then
asymptotically almost surely contains a cycle (and thus a path) of length at
least , and (ii) if , then
asymptotically almost surely contains a path of length at least . Our
theorems extend classical results on paths and cycles in the binomial random
graph, obtained by taking to be the complete graph on vertices.Comment: 26 page
Triangle-Intersecting Families of Graphs
A family of graphs F is said to be triangle-intersecting if for any two
graphs G,H in F, the intersection of G and H contains a triangle. A conjecture
of Simonovits and Sos from 1976 states that the largest triangle-intersecting
families of graphs on a fixed set of n vertices are those obtained by fixing a
specific triangle and taking all graphs containing it, resulting in a family of
size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations
(for example, we prove that the same is true of odd-cycle-intersecting
families, and we obtain best possible bounds on the size of the family under
different, not necessarily uniform, measures). We also obtain stability
results, showing that almost-largest triangle-intersecting families have
approximately the same structure.Comment: 43 page
Tight Hamilton cycles with high discrepancy
In this paper, we initiate the study of discrepancy questions for spanning
subgraphs of -uniform hypergraphs. Our main result is that any -colouring
of the edges of a -uniform -vertex hypergraph with minimum
-degree contains a tight Hamilton cycle with
high discrepancy, that is, with at least edges of one colour.
The minimum degree condition is asymptotically best possible and our theorem
also implies a corresponding result for perfect matchings. Our tools combine
various structural techniques such as Tur\'an-type problems and hypergraph
shadows with probabilistic techniques such as random walks and the nibble
method. We also propose several intriguing problems for future research.Comment: 20 pages, 1 figur
On rigidity, orientability and cores of random graphs with sliders
Suppose that you add rigid bars between points in the plane, and suppose that
a constant fraction of the points moves freely in the whole plane; the
remaining fraction is constrained to move on fixed lines called sliders. When
does a giant rigid cluster emerge? Under a genericity condition, the answer
only depends on the graph formed by the points (vertices) and the bars (edges).
We find for the random graph the threshold value of
for the appearance of a linear-sized rigid component as a function of ,
generalizing results of Kasiviswanathan et al. We show that this appearance of
a giant component undergoes a continuous transition for and a
discontinuous transition for . In our proofs, we introduce a
generalized notion of orientability interpolating between 1- and
2-orientability, of cores interpolating between 2-core and 3-core, and of
extended cores interpolating between 2+1-core and 3+2-core; we find the precise
expressions for the respective thresholds and the sizes of the different cores
above the threshold. In particular, this proves a conjecture of Kasiviswanathan
et al. about the size of the 3+2-core. We also derive some structural
properties of rigidity with sliders (matroid and decomposition into components)
which can be of independent interest.Comment: 32 pages, 1 figur
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