1,880 research outputs found
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
Ramsey numbers for partially-ordered sets
We present a refinement of Ramsey numbers by considering graphs with a
partial ordering on their vertices. This is a natural extension of the ordered
Ramsey numbers. We formalize situations in which we can use arbitrary families
of partially-ordered sets to form host graphs for Ramsey problems. We explore
connections to well studied Tur\'an-type problems in partially-ordered sets,
particularly those in the Boolean lattice. We find a strong difference between
Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the
partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl
Joint Routing and STDMA-based Scheduling to Minimize Delays in Grid Wireless Sensor Networks
In this report, we study the issue of delay optimization and energy
efficiency in grid wireless sensor networks (WSNs). We focus on STDMA (Spatial
Reuse TDMA)) scheduling, where a predefined cycle is repeated, and where each
node has fixed transmission opportunities during specific slots (defined by
colors). We assume a STDMA algorithm that takes advantage of the regularity of
grid topology to also provide a spatially periodic coloring ("tiling" of the
same color pattern). In this setting, the key challenges are: 1) minimizing the
average routing delay by ordering the slots in the cycle 2) being energy
efficient. Our work follows two directions: first, the baseline performance is
evaluated when nothing specific is done and the colors are randomly ordered in
the STDMA cycle. Then, we propose a solution, ORCHID that deliberately
constructs an efficient STDMA schedule. It proceeds in two steps. In the first
step, ORCHID starts form a colored grid and builds a hierarchical routing based
on these colors. In the second step, ORCHID builds a color ordering, by
considering jointly both routing and scheduling so as to ensure that any node
will reach a sink in a single STDMA cycle. We study the performance of these
solutions by means of simulations and modeling. Results show the excellent
performance of ORCHID in terms of delays and energy compared to a shortest path
routing that uses the delay as a heuristic. We also present the adaptation of
ORCHID to general networks under the SINR interference model
Semialgebraic Graphs having Countable List-Chromatic Numbers
The set of semialgebraic graphs having countable list-chromatic numbers is
characterized. Some other related sets of graphs having countable
list-chromatic numbers also are.Comment: This version has been completely rewritten. It will appear in PAM
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