1,422 research outputs found
Variations on Cops and Robbers
We consider several variants of the classical Cops and Robbers game. We treat
the version where the robber can move R > 1 edges at a time, establishing a
general upper bound of N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha
= 1 + 1/R, thus generalizing the best known upper bound for the classical case
R = 1 due to Lu and Peng. We also show that in this case, the cop number of an
N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N
if R is infinite. For R = 1, we study the directed graph version of the
problem, and show that the cop number of any strongly connected digraph on N
vertices is at most O(N(log log N)^2/log N). Our approach is based on
expansion.Comment: 18 page
Avoiding small subgraphs in Achlioptas processes
For a fixed integer r, consider the following random process. At each round,
one is presented with r random edges from the edge set of the complete graph on
n vertices, and is asked to choose one of them. The selected edges are
collected into a graph, which thus grows at the rate of one edge per round.
This is a natural generalization of what is known in the literature as an
Achlioptas process (the original version has r=2), which has been studied by
many researchers, mainly in the context of delaying or accelerating the
appearance of the giant component.
In this paper, we investigate the small subgraph problem for Achlioptas
processes. That is, given a fixed graph H, we study whether there is an online
algorithm that substantially delays or accelerates a typical appearance of H,
compared to its threshold of appearance in the random graph G(n, M). It is easy
to see that one cannot accelerate the appearance of any fixed graph by more
than the constant factor r, so we concentrate on the task of avoiding H. We
determine thresholds for the avoidance of all cycles C_t, cliques K_t, and
complete bipartite graphs K_{t,t}, in every Achlioptas process with parameter r
>= 2.Comment: 43 pages; reorganized and shortene
Packing tight Hamilton cycles in 3-uniform hypergraphs
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C
\subset H is a collection of N edges for which there is an ordering of the
vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i,
v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop
new techniques which enable us to prove that under certain natural
pseudo-random conditions, almost all edges of H can be covered by edge-disjoint
tight Hamilton cycles, for N divisible by 4. Consequently, we derive the
corollary that random 3-uniform hypergraphs can be almost completely packed
with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small.
Along the way, we develop a similar result for packing Hamilton cycles in
pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur
The random k-matching-free process
Let be a graph property which is preserved by removal of edges,
and consider the random graph process that starts with the empty -vertex
graph and then adds edges one-by-one, each chosen uniformly at random subject
to the constraint that is not violated. These types of random
processes have been the subject of extensive research over the last 20 years,
having striking applications in extremal combinatorics, and leading to the
discovery of important probabilistic tools. In this paper we consider the
-matching-free process, where is the property of not
containing a matching of size . We are able to analyse the behaviour of this
process for a wide range of values of ; in particular we prove that if
or if then this process is likely to
terminate in a -matching-free graph with the maximum possible number of
edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds
on are essentially best possible, and we make a first step towards
understanding the behaviour of the process in the intermediate regime
Ramsey games with giants
The classical result in the theory of random graphs, proved by Erdos and
Renyi in 1960, concerns the threshold for the appearance of the giant component
in the random graph process. We consider a variant of this problem, with a
Ramsey flavor. Now, each random edge that arrives in the sequence of rounds
must be colored with one of R colors. The goal can be either to create a giant
component in every color class, or alternatively, to avoid it in every color.
One can analyze the offline or online setting for this problem. In this paper,
we consider all these variants and provide nontrivial upper and lower bounds;
in certain cases (like online avoidance) the obtained bounds are asymptotically
tight.Comment: 29 pages; minor revision
Efficacy of needle-placement technique in radiofrequency ablation for treatment of lumbar facet arthropathy.
BACKGROUND:Many studies have assessed the efficacy of radiofrequency ablation to denervate the facet joint as an interventional means of treating axial low-back pain. In these studies, varying procedural techniques were utilized to ablate the nerves that innervate the facet joints. To date, no comparison studies have been performed to suggest superiority of one technique or even compare the prevalence of side effects and complications. MATERIALS AND METHODS:A retrospective chart review was performed on patients who underwent a lumbar facet denervation procedure. Each patient's chart was analyzed for treatment technique (early versus advanced Australian), preprocedural visual numeric scale (VNS) score, postprocedural VNS score, duration of pain relief, and complications. RESULTS:Pre- and postprocedural VNS scores and change in VNS score between the two groups showed no significant differences. Patient-reported benefit and duration of relief was greater in the advanced Australian technique group (P=0.012 and 0.022, respectively). The advanced Australian technique group demonstrated a significantly greater median duration of relief (4 months versus 1.5 months, P=0.022). Male sex and no pain-medication use at baseline were associated with decreased postablation VNS scores, while increasing age and higher preablation VNS scores were associated with increased postablation VNS scores. Despite increasing age being associated with increased postablation VNS scores, age and the advanced Australian technique were found to confer greater patient self-reported treatment benefit. CONCLUSION:The advanced Australian technique provides a significant benefit over the early Australian technique for the treatment of lumbar facet pain, both in magnitude and duration of pain relief
Packing Hamilton Cycles Online
It is known that w.h.p. the hitting time for the random
graph process to have minimum degree coincides with the hitting time
for edge disjoint Hamilton cycles. In this paper we prove an online
version of this property. We show that, for a fixed integer , if
random edges of are presented one by one then w.h.p. it is possible to
color the edges online with colors so that at time ,
each color class is Hamiltonian.Comment: Minor change
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