9,294 research outputs found
A Lower Bound for Radio -chromatic Number of an Arbitrary Graph
Radio -coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph , subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph with diameter and apositive integer , , a radio -coloring of is an assignment of positive integers to the vertices of such that , where and are any two distinct vertices of and is the distance between and .In this paper we give a lower bound for the radio -chromatic number of an arbitrarygraph in terms of , the total number of vertices and apositive integer such that for all . If is the triameter we get a better lower bound. We also find the triameter for several graphs, and show that the lower bound obtained for these graphs is sharp for the case
Improved Bounds for Radio k
A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (nonnegative integers) to the stations in an optimal way such that interference is avoided as reported by Hale (2005). Radio k-coloring of a graph is a special type of channel assignment problem. Kchikech et al. (2005) have given a lower and an upper bound for radio k-chromatic number of hypercube Qn, and an improvement of their lower bound was obtained by Kola and Panigrahi (2010). In this paper, we further improve Kola et al.'s lower bound as well as Kchikeck et al.'s upper bound. Also, our bounds agree for nearly antipodal number of Qn when n≡2 (mod 4)
Conflict-free coloring of graphs
We study the conflict-free chromatic number chi_{CF} of graphs from extremal
and probabilistic point of view. We resolve a question of Pach and Tardos about
the maximum conflict-free chromatic number an n-vertex graph can have. Our
construction is randomized. In relation to this we study the evolution of the
conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and
give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the
conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page
On hamiltonian colorings of block graphs
A hamiltonian coloring c of a graph G of order p is an assignment of colors
to the vertices of G such that for every two
distinct vertices u and v of G, where D(u,v) denoted the detour distance
between u and v. The value hc(c) of a hamiltonian coloring c is the maximum
color assigned to a vertex of G. The hamiltonian chromatic number, denoted by
hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this
paper, we present a lower bound for the hamiltonian chromatic number of block
graphs and give a sufficient condition to achieve the lower bound. We
characterize symmetric block graphs achieving this lower bound. We present two
algorithms for optimal hamiltonian coloring of symmetric block graphs.Comment: 12 pages, 1 figure. A conference version appeared in the proceedings
of WALCOM 201
Backbone colorings for networks: tree and path backbones
We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph and a spanning subgraph of (the backbone of ), a backbone coloring for and is a proper vertex coloring of in which the colors assigned to adjacent vertices in differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path
A general framework for coloring problems: old results, new results, and open problems
In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants
Coded Computation Against Processing Delays for Virtualized Cloud-Based Channel Decoding
The uplink of a cloud radio access network architecture is studied in which
decoding at the cloud takes place via network function virtualization on
commercial off-the-shelf servers. In order to mitigate the impact of straggling
decoders in this platform, a novel coding strategy is proposed, whereby the
cloud re-encodes the received frames via a linear code before distributing them
to the decoding processors. Transmission of a single frame is considered first,
and upper bounds on the resulting frame unavailability probability as a
function of the decoding latency are derived by assuming a binary symmetric
channel for uplink communications. Then, the analysis is extended to account
for random frame arrival times. In this case, the trade-off between average
decoding latency and the frame error rate is studied for two different queuing
policies, whereby the servers carry out per-frame decoding or continuous
decoding, respectively. Numerical examples demonstrate that the bounds are
useful tools for code design and that coding is instrumental in obtaining a
desirable compromise between decoding latency and reliability.Comment: 11 pages and 12 figures, Submitte
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