100,403 research outputs found
Distance-uniform graphs with large diameter
An ϵ-distance-uniform graph is one with a critical distance d such that from every vertex, all but at most an ϵ-fraction of the remaining vertices are at distance exactly d. Motivated by the theory of network creation games, Alon, Demaine, Hajiaghayi, and Leighton made the follow- ing conjecture of independent interest: that every ϵ-distance-uniform graph (and, in fact, a broader class of ϵ-distance-almost-uniform graphs) has critical distance at most logarithmic in the number of vertices n. We disprove this conjecture and characterize the asymptotics of this extremal prob- lem. Speci-cally, for 1/n ≤ ϵ ≤ 1 /log n , we construct ϵ-distance-uniform graphs with critical distance 2ω(log n/log ϵ-1). We also prove an upper bound on the critical distance of the form 2O(log n/log ϵ-1) for all ϵ and n. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest.Peer ReviewedPostprint (author's final draft
Decompositions into subgraphs of small diameter
We investigate decompositions of a graph into a small number of low diameter
subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E)
on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that
|E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the
subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma,
Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a
constant depending only \epsilon. This shows that every dense graph can be
partitioned into a small number of ``small worlds'' provided that few edges can
be ignored. Improving on their result, we determine P(n,\epsilon,d) within an
absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded
for \epsilon
n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We
also prove that if G has large minimum degree, all the edges of G can be
covered by a small number of low diameter subgraphs. Finally, we extend some of
these results to hypergraphs, improving earlier work of Polcyn, R\"odl,
Ruci\'nski, and Szemer\'edi.Comment: 18 page
Moving in temporal graphs with very sparse random availability of edges
In this work we consider temporal graphs, i.e. graphs, each edge of which is
assigned a set of discrete time-labels drawn from a set of integers. The labels
of an edge indicate the discrete moments in time at which the edge is
available. We also consider temporal paths in a temporal graph, i.e. paths
whose edges are assigned a strictly increasing sequence of labels. Furthermore,
we assume the uniform case (UNI-CASE), in which every edge of a graph is
assigned exactly one time label from a set of integers and the time labels
assigned to the edges of the graph are chosen randomly and independently, with
the selection following the uniform distribution. We call uniform random
temporal graphs the graphs that satisfy the UNI-CASE. We begin by deriving the
expected number of temporal paths of a given length in the uniform random
temporal clique. We define the term temporal distance of two vertices, which is
the arrival time, i.e. the time-label of the last edge, of the temporal path
that connects those vertices, which has the smallest arrival time amongst all
temporal paths that connect those vertices. We then propose and study two
statistical properties of temporal graphs. One is the maximum expected temporal
distance which is, as the term indicates, the maximum of all expected temporal
distances in the graph. The other one is the temporal diameter which, loosely
speaking, is the expectation of the maximum temporal distance in the graph. We
derive the maximum expected temporal distance of a uniform random temporal star
graph as well as an upper bound on both the maximum expected temporal distance
and the temporal diameter of the normalized version of the uniform random
temporal clique, in which the largest time-label available equals the number of
vertices. Finally, we provide an algorithm that solves an optimization problem
on a specific type of temporal (multi)graphs of two vertices.Comment: 30 page
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