937 research outputs found
A Review of Interference Reduction in Wireless Networks Using Graph Coloring Methods
The interference imposes a significant negative impact on the performance of
wireless networks. With the continuous deployment of larger and more
sophisticated wireless networks, reducing interference in such networks is
quickly being focused upon as a problem in today's world. In this paper we
analyze the interference reduction problem from a graph theoretical viewpoint.
A graph coloring methods are exploited to model the interference reduction
problem. However, additional constraints to graph coloring scenarios that
account for various networking conditions result in additional complexity to
standard graph coloring. This paper reviews a variety of algorithmic solutions
for specific network topologies.Comment: 10 pages, 5 figure
Recommended from our members
Edge-colourings of graphs
All the results in this thesis are concerned with the classification of graphs by their chromatic class.
We first extend earlier results of Fiorini and others to give a complete list of critical graphs of order at most ten. We give conditions for extending the edge-colouring of a nearly complete subgraph to the whole graph and use this result to prove a special case of Vizing's conjecture. We also use other methods to solve further cases of this conjecture.
A major part of the thesis classifies those graphs with at most 4 vertices of maximum degree and this work is generalised to graphs with r vertices of maximum degree. We also completely classify all regular graphs G with degree at least 6/7|V(G)|.
Finally we give some examples of even order critical graphs and introduce the concept of a supersnark
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric
Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications
Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable
Basic Neutrosophic Algebraic Structures and their Application to Fuzzy and Neutrosophic Models
The involvement of uncertainty of varying degrees when the total of the
membership degree exceeds one or less than one, then the newer mathematical
paradigm shift, Fuzzy Theory proves appropriate. For the past two or more
decades, Fuzzy Theory has become the potent tool to study and analyze
uncertainty involved in all problems. But, many real-world problems also abound
with the concept of indeterminacy. In this book, the new, powerful tool of
neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic
models are described. The theory of neutrosophic graphs is introduced and
applied to fuzzy and neutrosophic models. This book is organized into four
chapters. In Chapter One we introduce some of the basic neutrosophic algebraic
structures essential for the further development of the other chapters. Chapter
Two recalls basic graph theory definitions and results which has interested us
and for which we give the neutrosophic analogues. In this chapter we give the
application of graphs in fuzzy models. An entire section is devoted for this
purpose. Chapter Three introduces many new neutrosophic concepts in graphs and
applies it to the case of neutrosophic cognitive maps and neutrosophic
relational maps. The last section of this chapter clearly illustrates how the
neutrosophic graphs are utilized in the neutrosophic models. The final chapter
gives some problems about neutrosophic graphs which will make one understand
this new subject.Comment: 149 pages, 130 figure
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Edge colorings of graphs on surfaces and star edge colorings of sparse graphs
In my dissertation, I present results on two types of edge coloring problems for graphs.
For each surface Σ, we define ∆(Σ) = max{∆(G)| G is a class two graph with maximum degree ∆(G) that can be embedded in Σ}. Hence Vizing’s Planar Graph Conjecture can be restated as ∆(Σ) = 5 if Σ is a sphere. For a surface Σ with characteristic χ(Σ) ≤ 0, it is known ∆(Σ) ≥ H(χ(Σ))−1, where H(χ(Σ)) is the Heawood number of the surface, and if the Euler char- acteristic χ(Σ) ∈ {−7, −6, . . . , −1, 0}, ∆(Σ) is already known. I study critical graphs on general surfaces and show that (1) if G is a critical graph embeddable on a surface Σ with Euler character- istic χ(Σ) ∈ {−6, −7}, then ∆(Σ) = 10, and (2) if G is a critical graph embeddable on a surface Σ with Euler characteristic χ(Σ) ≤ −8, then ∆(G) ≤ H(χ(Σ)) (or H(χ(Σ))+1) for some special families of graphs, namely if the minimum degree is at most 11 or if ∆ is very large et al. As applications, we show that ∆(Σ) ≤ H (χ(Σ)) if χ(Σ) ∈ {−22, −21, −20, −18, −17, −15, . . . , −8}and ∆(Σ) ≤ H (χ(Σ)) + 1 if χ(Σ) ∈ {−53, . . . , 23, −19, −16}. Combining this with [19], it follows that if χ(Σ) = −12 and Σ is orientable, then ∆(Σ) = H(χ(Σ)).
A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored sub-
graph is a path of length at most 3. The star chromatic index χ′st(G) of a graph G is the smallest
integer k such that G has a star k-edge-coloring. The list star chromatic index ch′st(G) is defined
analogously. Bezegova et al. and Deng et al. independently proved that χ′ (T) ≤ 3∆ for anyst 2
tree T with maximum degree ∆. Here, we study the list star edge coloring and give tree-like
bounds for (list) star chromatic index of sparse graphs. We show that if mad(G) \u3c 2.4, then
χ′ (G)≤3∆+2andifmad(G)\u3c15,thench′ (G)≤3∆+1.Wealsoshowthatforeveryε\u3e0st 2 7 st 2
there exists a constant c(ε) such that if mad(G) \u3c 8 − ε, then ch′ (G) ≤ 3∆ + c(ε). We also3 st 2
find guaranteed substructures of graph with mad(G) \u3c 3∆ − ε which may be of interest in other2
problems for sparse graphs
- …