DOCSLIB.ORG

Comparing Graphs 69 5.1 Path and Cycle Comparison

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Comparing Graphs 69 5.1 Path and Cycle Comparison CALIFORNIA STATE UNIVERSITY SAN MARCOS THESIS SIGNATURE PAGE THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE MASTER OF SCIENCES IN MATHEMATICS THESIS TITLE: Communication Networks AUTHOR: Janina Silvana P. Patno DATE OF SUCCESSFUL DEFENSE: April 19,2011 THE THESIS HAS BEEN ACCEPTED BY THE THESIS COMMITTEE IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCES IN MATHEMATICS. Andre Ki.indgen t;;f~/kJt{ THESIS COMMITTEE CHAIR SIGNATA~ DliTE I K. Brooks Reid ~~~R~c& THESIS COMMITTEE MEMBER SIGNATURE ~IE Marshall WhittleseJ:: 1/JJtl!Jbt. ?(_'1/1} THESIS COMMITTEE MEMBER SIGNATURE ' DATE Thesis Abstract We consider the problem of sending a message in a communication network. Our networks are modeled by graphs. In our graphs we want to send a message from a vertex s, the sender, to a vertex r, the receiver, via paths from s tor. One difficulty we may encounter if we try sending the message along every possible s, r-path is that the receiver r will be flooded, so we focus our attention on certain finite protocols for sending the message from s to r. We study the probability that such a protocol successfully sends a message from s to r if the edges of the network live with probability p. We present some results and some open problems on this topic. Keywords: network, communication, protocol, two-terminal graph, two-terminal reliability polynomial ii Contents 1 Introduction 1 1.1 Introduction . 1 1.2 Basic Definitions 4 2 The Deterministic Model 11 2.1 s, r-Graphs . 11 2.2 Protocols for Message Transmission . 11 2.3 A Forbidden Minor Characterization 17 3 The Probabilistic Model 22 3.1 Reliability of a Protocol . 22 3.2 Reliability of the Complete Forwarding Protocol 26 3.3 Reliability and Deficiency of a Partial Forwarding Protocol . 38 4 Maximum Protocol Reliability 49 4.1 Protocols with Reliability p 49 4.2 Using Deficiencies to find p . 50 5 Comparing Graphs 69 5.1 Path and Cycle Comparison . 70 5.2 Comparing Polynomials . 75 5.3 Comparing the 10 Forbidden Minors 79 5.4 The Shortest s, r-Paths . 80 6 Kelmans' Type Construction 87 6.1 Definitions . 87 6.2 Operations With s, r-Graphs . 88 6.3 Multiplying Constructions . 94 6.4 Reliable Pairs with Prescribed Crossing Profiles 96 CONTENTS iii 7 A Graph with a Piecewise Polynomial for p 100 100 701 Graph B 0 With Edge Replaced 702 Open Problems 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 108 1 Chapter 1 Introduction 1.1 Introduction A network is an "interconnected system of things or people" [14] and networks, such as the internet, enable people to share information with one another. Com­ munication in a network has been studied by many researchers in many different forms and under different assumptions, but the goal is typically to determine the reliability of such networks. In 1959, during the Cold War, an engineer named Paul Baran was interested in survivable communications and felt that a more reliable system of communica­ tion would help prevent unnecessary attacks during the war. Baran's first attempt with such a communication system would try to send a "Go" or "No go" message from the president to commanders via AM radio. After three years of development, he was able to create a better communication system that would send longer mes­ sages with better survivability using the idea of packet switching and distributive communication in a network [1]. CHAPTER 1. INTRODUCTION 2 In this thesis we represent networks as graphs, where the vertices are the nodes or processors and the edges are links between processors. Thus, Baran's idea for a communication network would have edges and vertices that can fail indepen­ dently without recovering, and there would be multiple distinguished vertices, the terminals, that send and receive information. The model we study can be described with graphs with no multiple edges or loops such that the vertices do not fail, but the edges can fail independently without recovering. We focus our attention on the case where we have only two specified terminals, a sender and a receiver. Paraphrasing Douglas West (see [13], page 149), in order for a network to be able to reliably transmit information, it needs to stay connected even when some of the edges or vertices fail. So one way to study the reliability of a network is to study the number of walks with working edges between terminals. We study the reliability of a network Gin terms of the reliability polynomial, p(G,p), which is the probability that some walk remains between every two terminals if all edges can fail independently with probability p. One can study this connectivity question for any number of terminals k, where k ~ 2. So given a graph on n vertices one can look at the all terminal ( k = n) or k-terminal reliability polynomial. We will primarily study the two-terminal case. In 1992, Brown and Colbourn [4] calculated the all terminal reliability poly­ nomial for several families of graphs, such as trees and cycles, and studied the roots and coefficients of the reliability polynomial for a connected graph. Colbourn [6] CHAPTER 1. INTRODUCTION 3 recognized that constraints on a network force restrictions on the coefficients of the reliability polynomial and that these restrictions can be more noticeable by writing the reliability polynomial in different forms. Our study of communication in a network combines Baran's idea of packet switching in network transmission with the idea of a reliability polynomial to arrive at the concept of a protocol reliability polynomial. Like Colbourn, we calculate the reliability of specific networks and interpret our findings by writing the reliability polynomial in one or two variables. In Chapter 2, we review the deterministic model of the network as studied by Fich, Kiindgen, Pelsmajer and Ramamurthi [8, 9]. We describe our model as a graph with a fixed sender s and a fixed receiver r, and discuss how messages are sent in our model using protocols. We also describe the main result from [8], a forbidden minor characterization. In Chapter 3, we change the model described in Chapter 2, by attaching a probability to the existence of every edge. We study how well a protocol will work in our model by determining its reliability. Chapter 4 discusses which protocol will give us the best reliability for each of the 10 forbidden minors from Chapter 2. Chapter 5 compares graphs of paths and cycles to determine which would work better as a communication network. We also give a result about polynomials that can be used to compare the reliabilities of protocols in different graphs. In Chapter 6, we use an idea from Kelmans [10] to construct a pair of graphs such that if their reliability polynomials were graphed, they would cross a CHAPTER 1. INTRODUCTION 4 specified number of times. We end the thesis with Chapter 7, in which we use our findings of paths and cycles from Chapter 5 to show that there are graphs such that the best protocol to use at low probability is different from that at high probability. We present a number of fundamental results and some open problems on this topic. 1.2 Basic Definitions In this section we review some basic graph theory, probability, and algebra defini­ tions and theorems we will be using in this thesis. For a more comprehensive graph theory review please refer to West [13] and Diestel [7]. A comprehensive probability review can be found in Pitman [11]. A graph G consists of a non-empty finite vertex set V(G), and an edge multiset E(G), where each edge is an unordered pair of two distinct vertices called the endpoints. If vertex vis an endpoint of edge e, then v and e are incident. When u and v are the endpoints of an edge, they are adjacent and are neighbors, and we will denote this edge by uv. A complete graph is a graph whose vertices are pairwise adjacent. A subgraph H of G is a graph such that the V(H) ~ V(G) and E(H) C E(G). His said to be induced if E(H) consists of all edges in E(G) whose endpoints are both in V(H). A walk of length k is a finite sequence v0 , v1 , v2 , ... , Vk-l, vk of vertices such that vi-l vi, for all i with 1 :<::; i :<::; k, is an edge in G. A trail is a walk such that if vi = Vj for i =/= j, then vi+l =/= Vj+l· Thus, each edge uv can be used at most twice CHAPTER 1. INTRODUCTION 5 in a trail: once as u, v and once as v, u. A walk (trail) is closed if the endpoints, v0 and vk, of the walk (trail) are the same. A path is a walk Vo, v1, v2, ... , Vk-1, vk such that v0, v1, ... , vk-l, vk are distinct vertices of G. Given two specified vertices u and v, au, v-walk (u, v-trail, u, v-path) is a walk (trail, path) vo, v1, ... , vk that starts at v0 = u and ends at vk = v. The length of a path is its number of edges, k. A walk contains a path (trail) if the sequence of vertices describing the path (trail) is a subsequence of the sequence of vertices describing the walk, and every pair of consecutive vertices in the sequence describing path (trail) also occurs as a consecutive pair in the sequence describing the walk.
Load more

Recommended publications
  • Coloring ($ P 6 $, Diamond, $ K 4 $)-Free Graphs
    Coloring (P6, diamond, K4)-free graphs T. Karthick∗ Suchismita Mishra† June 28, 2021 Abstract We show that every (P6, diamond, K4)-free graph is 6-colorable. Moreover, we give an example of a (P6, diamond, K4)-free graph G with χ(G) = 6. This generalizes some known results in the literature. 1 Introduction We consider simple, finite, and undirected graphs. For notation and terminology not defined here we refer to [22]. Let Pn, Cn, Kn denote the induced path, induced cycle and complete graph on n vertices respectively. If G1 and G2 are two vertex disjoint graphs, then their union G1 ∪ G2 is the graph with V (G1 ∪ G2) = V (G1) ∪ V (G2) and E(G1 ∪ G2) = E(G1) ∪ E(G2). Similarly, their join G1 + G2 is the graph with V (G1 + G2) = V (G1) ∪ V (G2) and E(G1 + G2) = E(G1) ∪ E(G2)∪{(x,y) | x ∈ V (G1), y ∈ V (G2)}. For any positive integer k, kG denotes the union of k graphs each isomorphic to G. If F is a family of graphs, a graph G is said to be F-free if it contains no induced subgraph isomorphic to any member of F. A clique (independent set) in a graph G is a set of vertices that are pairwise adjacent (non-adjacent) in G. The clique number of G, denoted by ω(G), is the size of a maximum clique in G. A k-coloring of a graph G = (V, E) is a mapping f : V → {1, 2,...,k} such that f(u) 6= f(v) whenever uv ∈ E.
    [Show full text]
  • Isometric Diamond Subgraphs
    Isometric Diamond Subgraphs David Eppstein Computer Science Department, University of California, Irvine [email protected] Abstract. We test in polynomial time whether a graph embeds in a distance- preserving way into the hexagonal tiling, the three-dimensional diamond struc- ture, or analogous higher-dimensional structures. 1 Introduction Subgraphs of square or hexagonal tilings of the plane form nearly ideal graph draw- ings: their angular resolution is bounded, vertices have uniform spacing, all edges have unit length, and the area is at most quadratic in the number of vertices. For induced sub- graphs of these tilings, one can additionally determine the graph from its vertex set: two vertices are adjacent whenever they are mutual nearest neighbors. Unfortunately, these drawings are hard to find: it is NP-complete to test whether a graph is a subgraph of a square tiling [2], a planar nearest-neighbor graph, or a planar unit distance graph [5], and Eades and Whitesides’ logic engine technique can also be used to show the NP- completeness of determining whether a given graph is a subgraph of the hexagonal tiling or an induced subgraph of the square or hexagonal tilings. With stronger constraints on subgraphs of tilings, however, they are easier to con- struct: one can test efficiently whether a graph embeds isometrically onto the square tiling, or onto an integer grid of fixed or variable dimension [7]. In an isometric em- bedding, the unweighted distance between any two vertices in the graph equals the L1 distance of their placements in the grid. An isometric embedding must be an induced subgraph, but not all induced subgraphs are isometric.
    [Show full text]
  • Graphs with Few Trivial Characteristic Ideals
    University of Rhode Island DigitalCommons@URI Mathematics Faculty Publications Mathematics 2021 Graphs with few trivial characteristic ideals Carlos A. Alfaro Michael D. Barrus John Sinkovic Ralihe R. Villagrán Follow this and additional works at: https://digitalcommons.uri.edu/math_facpubs The University of Rhode Island Faculty have made this article openly available. Please let us know how Open Access to this research benefits you. This is a pre-publication author manuscript of the final, published article. Terms of Use This article is made available under the terms and conditions applicable towards Open Access Policy Articles, as set forth in our Terms of Use. Graphs with few trivial characteristic ideals Carlos A. Alfaroa, Michael D. Barrusb, John Sinkovicc and Ralihe R. Villagr´and a Banco de M´exico Mexico City, Mexico [email protected] bDepartment of Mathematics University of Rhode Island Kingston, RI 02881, USA [email protected] cDepartment of Mathematics Brigham Young University - Idaho Rexburg, ID 83460, USA [email protected] dDepartamento de Matem´aticas Centro de Investigaci´ony de Estudios Avanzados del IPN Apartado Postal 14-740, 07000 Mexico City, Mexico [email protected] Abstract We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1. We also give an alternative and simpler way to obtain the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1, and a list of minimal forbidden graphs for the family of graphs with Smith group having at most 4 invariant factors equal to 1.
    [Show full text]
  • Dynamical Systems Associated with Adjacency Matrices
    DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018190 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 5, July 2018 pp. 1945{1973 DYNAMICAL SYSTEMS ASSOCIATED WITH ADJACENCY MATRICES Delio Mugnolo Delio Mugnolo, Lehrgebiet Analysis, Fakult¨atMathematik und Informatik FernUniversit¨atin Hagen, D-58084 Hagen, Germany Abstract. We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss in detail qualitative properties of solutions to these problems by quadratic form methods. We distinguish between backward and forward evo- lution equations: the latter have typical features of diffusive processes, but cannot be well-posed on graphs with unbounded degree. On the contrary, well-posedness of backward equations is a typical feature of line graphs. We suggest how to detect even cycles and/or couples of odd cycles on graphs by studying backward equations for the adjacency matrix on their line graph. 1. Introduction. The aim of this paper is to discuss the properties of the linear dynamical system du (t) = Au(t) (1.1) dt where A is the adjacency matrix of a graph G with vertex set V and edge set E. Of course, in the case of finite graphs A is a bounded linear operator on the finite di- mensional Hilbert space CjVj. Hence the solution is given by the exponential matrix etA, but computing it is in general a hard task, since A carries little structure and can be a sparse or dense matrix, depending on the underlying graph.
    [Show full text]
  • Approximation Algorithms for Problems in Makespan Minimization on Unrelated Parallel Machines
    Western University Scholarship@Western Electronic Thesis and Dissertation Repository 4-8-2019 10:30 AM Approximation Algorithms for Problems in Makespan Minimization on Unrelated Parallel Machines Daniel R. Page The University of Western Ontario Supervisor Solis-Oba, Roberto The University of Western Ontario Graduate Program in Computer Science A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Daniel R. Page 2019 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Discrete Mathematics and Combinatorics Commons, and the Theory and Algorithms Commons Recommended Citation Page, Daniel R., "Approximation Algorithms for Problems in Makespan Minimization on Unrelated Parallel Machines" (2019). Electronic Thesis and Dissertation Repository. 6109. https://ir.lib.uwo.ca/etd/6109 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract A fundamental problem in scheduling is makespan minimization on unrelated parallel ma- chines (RjjCmax). Let there be a set J of jobs and a set M of parallel machines, where every + job J j 2 J has processing time or length pi; j 2 Q on machine Mi 2 M. The goal in RjjCmax is to schedule the jobs non-preemptively on the machines so as to minimize the length of the schedule, the makespan. A ρ-approximation algorithm produces in polynomial time a feasible solution such that its objective value is within a multiplicative factor ρ of the optimum, where ρ is called its approximation ratio.
    [Show full text]
  • Graph-Theoretic Properties of the Class of Phonological Neighbourhood Networks
    Graph-theoretic Properties of the Class of Phonological Neighbourhood Networks Rory Turnbull Newcastle University [email protected] Abstract flan clan This paper concerns the structure of phono- plant logical neighbourhood networks, which are a graph-theoretic representation of the phono- plane logical lexicon. These networks represent each word as a node and links are placed be- plan planner tween words which are phonological neigh- bours, usually defined as a string edit distance plaque of one. Phonological neighbourhood networks have been used to study many aspects of the planned mental lexicon and psycholinguistic theories pan of speech production and perception. This pa- plans per offers preliminary graph-theoretic observa- tions about phonological neighbourhood net- Figure 1: Example phonological neighbourhood net- works considered as a class. To aid this ex- work centred around the English word plan. Note that ploration, this paper introduces the concept of some neighbours of a word are neighbours of each the hyperlexicon, the network consisting of all other. Adapted from Turnbull and Peperkamp(2017). possible words for a given symbol set and their neighbourhood relations. The construction of the hyperlexicon is discussed, and basic prop- 0 0 erties are derived. This work is among the first of w), intransitive (if w is a neighbour of w , and w to directly address the nature of phonological is a neighbour of w00, it is not necessarily the case neighbourhood networks from an analytic per- that w is a neighbour of w00), and anti-reflexive (w spective. cannot be a neighbour of itself). Figure1 shows an abbreviated phonological 1 Motivation neighbourhood network for some words of English.
    [Show full text]
  • Gridline Graphs in Three and Higher Dimensions Susanna Lange Grand Valley State University
    Grand Valley State University ScholarWorks@GVSU Honors Projects Undergraduate Research and Creative Practice 12-2015 Gridline Graphs in Three and Higher Dimensions Susanna Lange Grand Valley State University Holly Raglow Grand Valley State University Follow this and additional works at: http://scholarworks.gvsu.edu/honorsprojects Part of the Physical Sciences and Mathematics Commons Recommended Citation Lange, Susanna and Raglow, Holly, "Gridline Graphs in Three and Higher Dimensions" (2015). Honors Projects. 535. http://scholarworks.gvsu.edu/honorsprojects/535 This Open Access is brought to you for free and open access by the Undergraduate Research and Creative Practice at ScholarWorks@GVSU. It has been accepted for inclusion in Honors Projects by an authorized administrator of ScholarWorks@GVSU. For more information, please contact [email protected]. Gridline Graphs in Three and Higher Dimensions Susanna Lange, Holly Raglow Advisor: Dr. Feryal Alayont December 16, 2015 Introduction In mathematics, a graph is a collection of vertices and edges represented by an image with vertices as points and edges as lines. While there are many different types of graphs in graph theory, we will focus on gridline graphs (also called graphs of (0; 1) matrices, or rook's graphs). Definition. In two dimensions, a gridline graph is a graph whose vertices can be labeled by points in the plane in such a way that two vertices are adjacent, meaning they share an edge, if and only if they have one coordinate in common. In other words, in a two dimensional gridline graph, two vertices are not connected if they differ in both the x coordinate and the y coordinate.
    [Show full text]
  • Algorithms for Generating Star and Path of Graphs Using BFS Dr
    Web Site: www.ijettcs.org Email: [email protected], [email protected] Volume 1, Issue 3, September – October 2012 ISSN 2278-6856 Algorithms for Generating Star and Path of Graphs using BFS Dr. H. B. Walikar2, Ravikumar H. Roogi1, Shreedevi V. Shindhe3, Ishwar. B4 2,4Prof. Dept of Computer Science, Karnatak University, Dharwad, 1,3Research Scholars, Dept of Computer Science, Karnatak University, Dharwad Abstract: In this paper we deal with BFS algorithm by 1.8 Diamond Graph: modifying it with some conditions and proper labeling of The diamond graph is the simple graph on nodes and vertices which results edges illustrated Fig.7. [2] and on applying it to some small basic 1.9 Paw Graph: class of graphs. The BFS algorithm has to modify The paw graph is the -pan graph, which is also accordingly. Some graphs will result in and by direct application of BFS where some need modifications isomorphic to the -tadpole graph. Fig.8 [2] in the algorithm. The BFS algorithm starts with a root vertex 1.10 Gem Graph: called start vertex. The resulted output tree structure will be The gem graph is the fan graph illustrated Fig.9 [2] in the form of Structure or in Structure. 1.11 Dart Graph: Keywords: BFS, Graph, Star, Path. The dart graph is the -vertex graph illustrated Fig.10.[2] 1.12 Tetrahedral Graph: 1. INTRODUCTION The tetrahedral graph is the Platonic graph that is the 1.1 Graph: unique polyhedral graph on four nodes which is also the A graph is a finite collection of objects called vertices complete graph and therefore also the wheel graph .
    [Show full text]
  • Graceful Labeling on Twig Diamond Graph with Pendant Edges ∗
    Bull. Pure Appl. Sci. Sect. E Math. Stat. 39E(2), 188–192 (2020) e-ISSN:2320-3226, Print ISSN:0970-6577 Bulletin of Pure and Applied Sciences DOI: 10.5958/2320-3226.2020.00018.1 Section - E - Mathematics & Statistics ©Dr. A.K. Sharma, BPAS PUBLICATIONS, 115-RPS- DDA Flat, Mansarover Park, Website : https : ==www:bpasjournals:com= Shahdara, Delhi-110032, India. 2020 Graceful labeling on twig diamond graph with pendant edges ∗ J. Jeba Jesintha1;y, Subashini K.2 and Allu Merin Sabu3 1,3. P.G. Department of Mathematics, Women’s Christian College, Chennai, Tamil Nadu, India. 2. Department of Mathematics, Jeppiaar Engineering College, Chennai, Tamil Nadu, India. 2. Research Scholar (Part-Time), P.G. Department of Mathematics, Women’s Christian College, Affiliated to University of Madras, Chennai, Tamil Nadu, India. 1. E-mail: [email protected] 2. E-mail: [email protected] , 3. E-mail: [email protected] Abstract A graceful labeling of a graph G with q edges is an injection f : V (G) ! f0; 1; 2; : : : ; qg with the property that the resulting edges are also distinct, where an edge incident with the vertices u and v is assigned the label jf(u) − f(v)j. A graph which admits a graceful labeling is called a graceful graph. In this paper, we prove the graceful labeling of a new family of graphs G called a twig diamond graph with pendant edges. Key words Graceful labeling, star graph, diamond graph. 2020 Mathematics Subject Classification 05C78. 1 Introduction The most interesting and famous graph labeling method is the graceful labeling of graphs introduced by Rosa [4] in 1967.
    [Show full text]
  • Quick but Odd Growth of Cacti
    Algorithmica (2017) 79:271–290 DOI 10.1007/s00453-017-0317-1 Quick but Odd Growth of Cacti Sudeshna Kolay1 · Daniel Lokshtanov2 · Fahad Panolan2 · Saket Saurabh1 Received: 31 December 2015 / Accepted: 25 April 2017 / Published online: 8 May 2017 © Springer Science+Business Media New York 2017 Abstract Let F be a family of graphs. Given an n-vertex input graph G and a positive integer k, testing whether G has a vertex subset S of size at most k, such that G − S belongs to F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when F is either the family of forests of cacti or the family of forests of odd-cacti. A graph H is called a forest of cacti if every pair of cycles in H intersect on at most one vertex. Furthermore, a forest of cacti H is called a forest of odd cacti, if every cycle of H is of odd length. Let us denote by C and Codd, the families of forests of cacti and forests of odd cacti, respectively. The vertex deletion problems corresponding to C and Codd are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with worst case run time 12knO(1) for both these problems. Our algorithms The conference version of this paper has been published in the Proceedings of IPEC 2015. The research leading to these results has received funding from the European Research Council (ERC) via grants Rigorous Theory of Preprocessing, reference 267959 and PARAPPROX, reference 306992 and the Bergen Research Foundation via grant “Beating Hardness by Preprocessing”.
    [Show full text]
  • Harmonious Coloring of Central Graphs of Certain Snake Graphs 571
    Applied Mathematical Sciences, Vol. 9, 2015, no. 12, 569 - 578 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121012 Harmonious Coloring of Central Graphs of Certain Snake Graphs M. S. Franklin Thamil Selvi Sathyabama University, Chennai, India Copyright © 2014 M. S. Franklin Thamil Selvi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A harmonious coloring of a simple graph is the proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number of G, denoted by χh (G), is the least number of colors in a harmonious coloring of G. The paper gives the structural properties and the estimates of harmonious chromatic number of central graphs of Triangular snake graph C [Tn], Double triangular snake graph C [D (Tn)] and Diamond snake graph C [Dn]. Keywords: Harmonious coloring, Central graph, Triangular snake, Double triangular snake, Diamond snake 1. Introduction Graph Theory is one of the most developing branches of mathematics with wide applications to computer science. Graph Theory is applied in diverse areas such as social sciences, linguistics, physical sciences, communication engineering and others. Graph coloring is one of the oldest and an interesting problem that comes up in lots of applications. Graph coloring is an assignment of colors (or any distinct marks) to the vertices of a graph. A vertex coloring is called proper coloring if no two adjacent vertices of the graph receive the same color and the graph is then called properly colored graph.
    [Show full text]
  • On the Purity of Minor-Closed Classes of Graphs
    On the purity of minor-closed classes of graphs Colin McDiarmid1 and Micha lPrzykucki2 1Department of Statistics, Oxford University∗ 2Mathematical Institute, Oxford University† August 31, 2016 Abstract Given a graph H with at least one edge, let gapH (n) denote the maximum difference between the numbers of edges in two n-vertex edge-maximal graphs with no minor H. We show that for exactly four connected graphs H (with at least two vertices), the class of graphs with no minor H is pure, that is, gapH (n) = 0 for all n > 1; and for each connected graph H (with at least two vertices) we have the dichotomy that either gapH (n) = O(1) or gapH (n) = Θ(n). Further, if H is 2-connected and does not yield a pure class, then there is a constant c > 0 such that gapH (n) ∼ cn. We also give some partial results when H is not connected or when there are two or more excluded minors. 1 Introduction We say that a graph G contains a graph H as a minor if we can obtain a graph isomorphic to H from a subgraph of G by using edge contractions (discarding any loops and multiple edges, we are interested in simple graphs). A class of graphs A is minor-closed if for each G 2 A, each minor G0 of G is also in A. We say that H is an excluded minor for A if H is not in A but each minor of H (other than H itself) is in A. Robertson and Seymour [16] showed that, for each minor-closed class A of graphs, the set H of excluded minors is finite.
    [Show full text]