108 research outputs found
Correlation among runners and some results on the Lonely Runner Conjecture
The Lonely Runner Conjecture was posed independently by Wills and Cusick and
has many applications in different mathematical fields, such as diophantine
approximation. This well-known conjecture states that for any set of runners
running along the unit circle with constant different speeds and starting at
the same point, there is a moment where all of them are far enough from the
origin. We study the correlation among the time that runners spend close to the
origin. By means of these correlations, we improve a result of Chen on the gap
of loneliness and we extend an invisible runner result of Czerwinski and
Grytczuk. In the last part, we introduce dynamic interval graphs to deal with a
weak version of the conjecture thus providing some new results.Comment: 18 page
Matchings in Random Biregular Bipartite Graphs
We study the existence of perfect matchings in suitably chosen induced
subgraphs of random biregular bipartite graphs. We prove a result similar to a
classical theorem of Erdos and Renyi about perfect matchings in random
bipartite graphs. We also present an application to commutative graphs, a class
of graphs that are featured in additive number theory.Comment: 30 pages and 3 figures - Latest version has updated introduction and
bibliograph
Bounds for identifying codes in terms of degree parameters
An identifying code is a subset of vertices of a graph such that each vertex
is uniquely determined by its neighbourhood within the identifying code. If
\M(G) denotes the minimum size of an identifying code of a graph , it was
conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there
exists a constant such that if a connected graph with vertices and
maximum degree admits an identifying code, then \M(G)\leq
n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any ,
\M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs
containing, among others, all regular graphs and all graphs of bounded clique
number. This settles the conjecture (up to constants) for these classes of
graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}.
In a second part, we prove that in any graph of minimum degree and
girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n.
Using the former result, we give sharp estimates for the size of the minimum
identifying code of random -regular graphs, which is about
Critical percolation on random regular graphs
We show that for all the size of the largest
component of a random -regular graph on vertices around the percolation
threshold is , with high probability. This extends
known results for fixed and for , confirming a prediction of
Nachmias and Peres on a question of Benjamini. As a corollary, for the largest
component of the percolated random -regular graph, we also determine the
diameter and the mixing time of the lazy random walk. In contrast to previous
approaches, our proof is based on a simple application of the switching method.Comment: 10 page
Decomposition of bounded degree graphs into -free subgraphs
We prove that every graph with maximum degree admits a partition of
its edges into parts (as ) none of which
contains as a subgraph. This bound is sharp up to a constant factor. Our
proof uses an iterated random colouring procedure.Comment: 8 pages; to appear in European Journal of Combinatoric
Random subgraphs make identification affordable
An identifying code of a graph is a dominating set which uniquely determines
all the vertices by their neighborhood within the code. Whereas graphs with
large minimum degree have small domination number, this is not the case for the
identifying code number (the size of a smallest identifying code), which indeed
is not even a monotone parameter with respect to graph inclusion.
We show that every graph with vertices, maximum degree
and minimum degree , for some
constant , contains a large spanning subgraph which admits an identifying
code with size . In particular, if
, then has a dense spanning subgraph with identifying
code , namely, of asymptotically optimal size. The
subgraph we build is created using a probabilistic approach, and we use an
interplay of various random methods to analyze it. Moreover we show that the
result is essentially best possible, both in terms of the number of deleted
edges and the size of the identifying code
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