25 research outputs found
On the chromatic number of the preferential attachment graph
We prove that for every and every , the
chromatic number of the preferential attachment graph is
asymptotically almost surely equal to . The proof relies on a
combinatorial construction of a family of digraphs of chromatic number
followed by a proof that asymptotically almost surely there is a digraph in
this family, which is realised as a subgraph of the preferential attachment
graph.Comment: 14 pages, 3 figure
New results for the random nearest neighbor tree
In this paper, we study the online nearest neighbor random tree in dimension
(called -NN tree for short) defined as follows. We fix the
torus of dimension and area and equip it with the
metric inherited from the Euclidean metric in . Then, embed
consecutively vertices in uniformly at random and
independently, and let each vertex but the first one connect to its (already
embedded) nearest neighbor. Call the resulting graph .
We show multiple results concerning the degree sequence of . First, we
prove that typically the number of vertices of degree at least
in the -NN tree decreases exponentially with and is tightly concentrated
by a new Lipschitz-type concentration inequality that may be of independent
interest. Second, we obtain that the maximum degree of is of logarithmic
order. Third, we give explicit bounds for the number of leaves that are
independent of the dimension and also give estimates for the number of paths of
length two. Moreover, we show that typically the height of a uniformly chosen
vertex in is and the diameter of is
, independently of the dimension.
Finally, we define a natural infinite analog of and show
that it corresponds to the local limit of the sequence of finite graphs
. Moreover, we prove almost surely that is
locally finite, that the simple random walk on is recurrent, and
that is connected
On the minimum bisection of random regular graphs
In this paper we give new asymptotically almost sure lower and upper bounds
on the bisection width of random regular graphs. The main contribution is a
new lower bound on the bisection width of , based on a first moment
method together with a structural decomposition of the graph, thereby improving
a 27 year old result of Kostochka and Melnikov. We also give a complementary
upper bound of , combining known spectral ideas with original
combinatorial insights. Developping further this approach, with the help of
Monte Carlo simulations, we obtain a non-rigorous upper bound of .Comment: 48 pages, 20 figure
The maximal running time of hypergraph bootstrap percolation
We show that for every , the maximal running time of the
-bootstrap percolation in the complete -uniform hypergraph on
vertices is . This answers a recent question of Noel
and Ranganathan in the affirmative, and disproves a conjecture of theirs.
Moreover, we show that the prefactor is of the form
as .Comment: 10 pages, 2 figures, improved presentatio
On the modularity of 3-regular random graphs and random graphs with given degree sequences
The modularity of a graph is a parameter introduced by Newman and Girvan
measuring its community structure; the higher its value (between and ),
the more clustered a graph is.
In this paper we show that the modularity of a random regular graph is at
least asymptotically almost surely (a.a.s.), thereby proving a
conjecture of McDiarmid and Skerman stating that a random regular graph has
modularity strictly larger than a.a.s. We also improve the upper
bound given therein by showing that the modularity of such a graph is a.a.s. at
most .
For a uniformly chosen graph over a given bounded degree sequence with
average degree and with many connected components, we
distinguish two regimes with respect to the existence of a giant component. In
more detail, we precisely compute the second term of the modularity in the
subcritical regime. In the supercritical regime, we further prove that there is
depending on the degree sequence, for which the modularity is
a.a.s. at least \begin{equation*} \dfrac{2\left(1 -
\mu\right)}{d(G_n)}+\varepsilon, \end{equation*} where is the
asymptotically almost sure limit of .Comment: 41 page
Local dimer dynamics in higher dimensions
We consider local dynamics of the dimer model (perfect matchings) on
hypercubic boxes . These consist of successively switching the dimers
along alternating cycles of prescribed (small) lengths. We study the
connectivity properties of the dimer configuration space equipped with these
transitions. Answering a question of Freire, Klivans, Milet and Saldanha, we
show that in three dimensions any configuration admits an alternating cycle of
length at most 6. We further establish that any configuration on
features order alternating cycles of length at most . We also
prove that the dynamics of dimer configurations on the unit hypercube of
dimension is ergodic when switching alternating cycles of length at most
. Finally, in the planar but non-bipartite case, we show that
parallelogram-shaped boxes in the triangular lattice are ergodic for switching
alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon
and R\'emila, which also uses 8-cycles. None of our proofs make reference to
height functions.Comment: 14 pages, 4 figure
Localization Game for Random Geometric Graphs
The localization game is a two player combinatorial game played on a graph
. The cops choose a set of vertices with .
The robber then chooses a vertex whose location is hidden from the
cops, but the cops learn the graph distance between the current position of the
robber and the vertices in . If this information is sufficient to locate
the robber, the cops win immediately; otherwise the cops choose another set of
vertices with , and the robber may move to a
neighbouring vertex. The new distances are presented to the robber, and if the
cops can deduce the new location of the robber based on all information they
accumulated thus far, then they win; otherwise, a new round begins. If the
robber has a strategy to avoid being captured, then she wins. The localization
number is defined to be the smallest integer so that the cops win the game.
In this paper we determine the localization number (up to poly-logarithmic
factors) of the random geometric graph slightly above
the connectivity threshold
The jump of the clique chromatic number of random graphs
The clique chromatic number of a graph is the smallest number of colors in a
vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid,
Mitsche and Pralat noted that around p \approx n^{-1/2} the clique chromatic
number of the random graph G_{n,p} changes by n^{\Omega(1)} when we increase
the edge-probability p by n^{o(1)}, but left the details of this surprising
phenomenon as an open problem.
We settle this problem, i.e., resolve the nature of this polynomial `jump' of
the clique chromatic number of the random graph G_{n,p} around edge-probability
p \approx n^{-1/2}. Our proof uses a mix of approximation and concentration
arguments, which enables us to (i) go beyond Janson's inequality used in
previous work and (ii) determine the clique chromatic number of G_{n,p} up to
logarithmic factors for any edge-probability p.Comment: 14 page