1,864 research outputs found
Tight Bounds on The Clique Chromatic Number
The clique chromatic number of a graph is the minimum number of colours
needed to colour its vertices so that no inclusion-wise maximal clique which is
not an isolated vertex is monochromatic. We show that every graph of maximum
degree has clique chromatic number
. We obtain as a corollary that every
-vertex graph has clique chromatic number . Both these results are tight
On Throughput and Decoding Delay Performance of Instantly Decodable Network Coding
In this paper, a comprehensive study of packet-based instantly decodable
network coding (IDNC) for single-hop wireless broadcast is presented. The
optimal IDNC solution in terms of throughput is proposed and its packet
decoding delay performance is investigated. Lower and upper bounds on the
achievable throughput and decoding delay performance of IDNC are derived and
assessed through extensive simulations. Furthermore, the impact of receivers'
feedback frequency on the performance of IDNC is studied and optimal IDNC
solutions are proposed for scenarios where receivers' feedback is only
available after and IDNC round, composed of several coded transmissions.
However, since finding these IDNC optimal solutions is computational complex,
we further propose simple yet efficient heuristic IDNC algorithms. The impact
of system settings and parameters such as channel erasure probability, feedback
frequency, and the number of receivers is also investigated and simple
guidelines for practical implementations of IDNC are proposed.Comment: This is a 14-page paper submitted to IEEE/ACM Transaction on
Networking. arXiv admin note: text overlap with arXiv:1208.238
Some results on chromatic number as a function of triangle count
A variety of powerful extremal results have been shown for the chromatic
number of triangle-free graphs. Three noteworthy bounds are in terms of the
number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994),
and Johansson. There have been comparatively fewer works extending these types
of bounds to graphs with a small number of triangles. One noteworthy exception
is a result of Alon et. al (1999) bounding the chromatic number for graphs with
low degree and few triangles per vertex; this bound is nearly the same as for
triangle-free graphs. This type of parametrization is much less rigid, and has
appeared in dozens of combinatorial constructions.
In this paper, we show a similar type of result for as a function
of the number of vertices , the number of edges , as well as the triangle
count (both local and global measures). Our results smoothly interpolate
between the generic bounds true for all graphs and bounds for triangle-free
graphs. Our results are tight for most of these cases; we show how an open
problem regarding fractional chromatic number and degeneracy in triangle-free
graphs can resolve the small remaining gap in our bounds
Dependent Random Graphs and Multiparty Pointer Jumping
We initiate a study of a relaxed version of the standard Erdos-Renyi random
graph model, where each edge may depend on a few other edges. We call such
graphs "dependent random graphs". Our main result in this direction is a
thorough understanding of the clique number of dependent random graphs. We also
obtain bounds for the chromatic number. Surprisingly, many of the standard
properties of random graphs also hold in this relaxed setting. We show that
with high probability, a dependent random graph will contain a clique of size
, and the chromatic number will be at most
.
As an application and second main result, we give a new communication
protocol for the k-player Multiparty Pointer Jumping (MPJ_k) problem in the
number-on-the-forehead (NOF) model. Multiparty Pointer Jumping is one of the
canonical NOF communication problems, yet even for three players, its
communication complexity is not well understood. Our protocol for MPJ_3 costs
communication, improving on a bound of Brody
and Chakrabarti [BC08]. We extend our protocol to the non-Boolean pointer
jumping problem , achieving an upper bound which is o(n) for
any players. This is the first o(n) bound for and
improves on a bound of Damm, Jukna, and Sgall [DJS98] which has stood for
almost twenty years.Comment: 18 page
Sum Coloring : New upper bounds for the chromatic strength
The Minimum Sum Coloring Problem (MSCP) is derived from the Graph Coloring
Problem (GCP) by associating a weight to each color. The aim of MSCP is to find
a coloring solution of a graph such that the sum of color weights is minimum.
MSCP has important applications in fields such as scheduling and VLSI design.
We propose in this paper new upper bounds of the chromatic strength, i.e. the
minimum number of colors in an optimal solution of MSCP, based on an
abstraction of all possible colorings of a graph called motif. Experimental
results on standard benchmarks show that our new bounds are significantly
tighter than the previous bounds in general, allowing to reduce substantially
the search space when solving MSCP .Comment: pre-prin
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