An Algorithm for Drawing Planar Graphs

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Triple Connected Line Domination Number for Some Standard and Special Graphs

International Journal of Advanced Scientific Research and Management, Volume 3 Issue 8, Aug 2018 www.ijasrm.com ISSN 2455-6378 Triple Connected Line Domination Number For Some Standard And Special Graphs A.Robina Tony1, Dr. A.Punitha Tharani2 1 Research Scholar (Register Number: 12519), Department of Mathematics, St.Mary’s College(Autonomous), Thoothukudi – 628 001, Tamil Nadu, India, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli – 627 012, Tamil Nadu, India 2 Associate Professor, Department of Mathematics, St.Mary’s College(Autonomous), Thoothukudi – 628 001, Tamil Nadu, India, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, India Abstract vertex domination, the concept of edge The concept of triple connected graphs with real life domination was introduced by Mitchell and application was introduced by considering the Hedetniemi. A set F of edges in a graph G is existence of a path containing any three vertices of a called an edge dominating set if every edge in E graph G. A subset S of V of a non - trivial graph G is – F is adjacent to at least one edge in F. The said to be a triple connected dominating set, if S is a dominating set and the induced subgraph <S> is edge domination number γ '(G) of G is the triple connected. The minimum cardinality taken minimum cardinality of an edge dominating set over all triple connected dominating set is called the of G. An edge dominating set F of a graph G is triple connected domination number and is denoted a connected edge dominating set if the edge by γtc(G).
Minimal K-Connected Non-Hamiltonian Graphs

Graphs and Combinatorics (2018) 34:289–312 https://doi.org/10.1007/s00373-018-1874-z Minimal k-Connected Non-Hamiltonian Graphs Guoli Ding1 · Emily Marshall2 Received: 24 April 2017 / Revised: 31 January 2018 / Published online: 14 February 2018 © Springer Japan KK, part of Springer Nature 2018 Abstract In this paper, we explore minimal k-connected non-Hamiltonian graphs. Graphs are said to be minimal in the context of some containment relation; we focus on subgraphs, induced subgraphs, minors, and induced minors. When k = 2, we discuss all minimal 2-connected non-Hamiltonian graphs for each of these four rela- tions. When k = 3, we conjecture a set of minimal non-Hamiltonian graphs for the minor relation and we prove one case of this conjecture. In particular, we prove all 3-connected planar triangulations which do not contain the Herschel graph as a minor are Hamiltonian. Keywords Hamilton cycles · Graph minors 1 Introduction Hamilton cycles in graphs are cycles which visit every vertex of the graph. Determining their existence in a graph is an NP-complete problem and as such, there is a large body of research proving necessary and sufﬁcient conditions. In this paper, we analyze non-Hamiltonian graphs. In particular, we consider the following general question: The ﬁrst author was supported in part by NSF Grant DMS-1500699. The work for this paper was largely done while the second author was at Louisiana State University. B Emily Marshall marshalle@arcadia.edu Guoli Ding ding@math.lsu.edu 1 Mathematics Department, Louisiana State University, Baton Rouge, LA 70803, USA 2 Computer Science and Mathematics Department, Arcadia University, Glenside, PA 19038, USA 123 290 Graphs and Combinatorics (2018) 34:289–312 what are the minimal k-connected non-Hamiltonian graphs? A graph is minimal if it does not contain a smaller graph with the same properties.
Packing by Edge Disjoint Trees

Global Journal of Mathematical Sciences: Theory and Practical. ISSN 0974-3200 Volume 5, Number 4 (2013), pp. 245_250 © International Research Publication House http://www.irphouse.com Packing by Edge Disjoint Trees Elachini V. Lal1 and Karakkattu S. Parvathy2 1Dept. of Higher Secondary, Govt. HSS Kuttippuram, Kerala INDIA– 679571. lamathematics@gmail.com 2Dept. of Mathematics, St. Mary’s College, Thrissur, Kerala INDIA – 680020. parvathymaths@gmail.com Abstract A Graph G is said to be a packing by edge disjoint trees, if its edges can be partitioned into trees. In this paper, we discuss the packability of graphs of various types by edge disjoint tree and find its number w (edt G) and its least number wl (edt G) Key Words: Packing, Edge disjoint trees. 1. Introduction: By a graph G = (V, E) we mean a finite, undirected, connected graph with no loop or multiple edges. The order and size of G are denoted by ‘m’ and ‘n’ respectively. For basic graph theoretic terminology, we refer to [1, 2, 3]. The concept of random packability was introduced by Sergio Ruiz [4] under the name of ‘random decomposable graphs’. Ruiz [4] obtained a characterization of randomly F- packable graphs, when F is P3 or K2. Lowell W Beineke [5] and Peter Hamburger [5] and Wayne D Goddard [5] characterized F-packable graphs where F is Kn, P4, P5, or P6. S.Arumugham [6] and S.Meena [6] extended a characterization of the random packability of two or more disconnected graphs like Kn U K1, n, C4 U P2 ,3K2 .Elachini V. Lal [7] and Karakkattu S.
Spindown Polyhedra

Spindown Polyhedra ANTHONY F. CONSTANTINIDES and GEORGE A. CONSTANTINIDES ARTICLE HISTORY Compiled March 20, 2018 1. Introduction Magic: The Gathering is a trading card game published by Wizards of the Coast [1]. The aim of most variants of the game is to reduce your opponent's life total from twenty to zero, thus winning the game. As it may take several turns to reduce a player's life total to zero, players need a mechanism to keep track of their current life total. For this purpose, players often use a device called a spindown life counter, shown in Figure 1. A spindown life counter is an icosahedron with a special labelling of the faces, such that - starting with 20 life total - a player can reduce their life total in decrements of one by rolling the icosahedron onto an adjacent face each time. A spindown life counter appears similar to a standard icosahedral die, known in gaming as a d20, however, the labelling of faces is diﬀerent, as also shown in Figure 1. Figure 1. Left: A spindown life counter from Magic: The Gathering. Right: A standard icosahedral die (d20). The ﬁrst author posed the question of whether it is possible to construct other polyhedra having a similar `spindown' property. Some simple experimentation revealed that is it possible to re-label the faces of a standard six-faced die (d6) to produce a spindown cube (see Figure 2); further experimentation revealed this to be the case for all Platonic solids. It can also be seen that it is possible, in all these cases, to chose the labelling such that it is not only spindown but also the face with the lowest label adjacent to the face with the highest label.
On the Euclidean Dimension of Graphs

ON THE EUCLIDEAN DIMENSION OF GRAPHS JIN HYUP HONG GREAT NECK SOUTH HIGH SCHOOL, GREAT NECK, NY AND DAN ISMAILESCU MATHEMATICS DEPARTMENT, HOFSTRA UNIVERSITY, NY arXiv:1501.00204v1 [math.MG] 31 Dec 2014 1 Abstract. The Euclidean dimension a graph G is defined to be the smallest integer d such that the vertices of G can be located in Rd in such a way that two vertices are unit distance apart if and only if they are adjacent in G. In this paper we determine the Euclidean dimension for twelve well known graphs. Five of these graphs, Dürer, Franklin, Desargues, Heawood and Tietze can be embedded in the plane, while the remaining graphs, Chvátal, Goldner-Harrary, Herschel, Fritsch, Grötzsch, Hoffman and Soifer have Euclidean dimension 3. We also present explicit embeddings for all these graphs. 1. History and previous work The Euclidean dimension of a graph G = (V, E), denoted dim(G) is the least integer n such that there exists a 1 : 1 embedding f : V Rn for which f(u) f(v) = 1 if and only → | − | if uv E. ∈ The concept was introduced by Erd˝os, Harary and Tutte in their seminal paper [7], where the authors determine the Euclidean dimension for several classes of graphs. For instance, they show that dim(K ) = n 1, where K is the complete graph on n n − n vertices. Using a construction due to Lenz, they also compute the Euclidean dimension of Km,n, the complete bipartite graph with m vertices in one class and n vertices in the other.
4 Euler Tours and Hamilton Cycles

4 Euler Tours and Hamilton Cycles 4.1 EULER TOURS A trail that traverses every edge of G is called an Euler trail of G because Euler was the first to investigate the existence of such trails in graphs. In the earliest known paper on graph theory (Euler, 1736), he showed that it was impossible to cross each of the seven bridges of Konigsberg once and only once during a walk through the town. A plan of KOlligsberg "and the river Pregel is shown in figure 4.1 a. As can be seen, proving that such a walk is impossible amounts to showing that the graph" of figure 4.1 b contains no Euler trail. A tour of G is a ~losed walk that traverses each edge of G at least once. An Euler tour is a tour which traverses each edge exactly once (in other words, a closed Euler trail). A graph is eulerian if it contains an Euler tour. l-'heorem 4.1 A nonempty connected· graph is eulerian if and only if it has no vertices of odd degree. Proof Let G be eulerian, and let C be an Euler tour ~f G with origin (and terminus) u. Each time a vertex v ~ccurs as an internal vertex of C, two of the edges incident with v are accounted for~ Since an Euler tour contains c A LJ---------~iilUB o (0) ( b) Figure 4.1. The bridges of Konigsberg and their graph 52 ' Graph Theory with Applications every edge of G, d(v) is even for all v ¢ u.
(V, E ) Be a Graph and Let F Be a Function That Assigns to Each Vertex of F:V(G) → {1,2,.....K} Such That for V to a Set of Values from the Set {1,2

BALANCED DOMINATION NUMBER OF SPECIAL GRAPHS 1S.CHRISTILDA and 2P.NAMASIVAYAM 1Department of Mathematics, Sadakathullah Appa College, Tirunelveli – 627011, Tamil Nadu, INDIA. E-mail: christilda2001@gmail.com 2PG and Research Department of Mathematics, The MDT Hindu College, Tirunelveli – 627010, Tamil Nadu, INDIA. ABSTRACT INTRODUCTION Let G= (V, E) be a graph. A Subset D of V is called a dominating Let G = (V, E ) be a graph and let f set of G if every vertex in V-D is be a function that assigns to each adjacent to atleast one vertex in D. The Domination number γ (G) of G is the vertex of V to a set of values from cardinality of the minimum dominating set of G. Let the set {1,2,.......k} that is, G = (V, E ) be a graph and let f be a function that assigns to each vertex of f:V(G) → {1,2,.....k} such that for V to a set of values from the set {1,2,.......k} that is, f:V(G) → each u,v ϵ V(G), f(u)≠f(v), if u is {1,2,.....k} such that for each u,v V(G), f(u) ≠ f(v), if u is adjacent to v in adjacent to v in G. Then the set D G. Then the dominating set D V (G) V (G) is called a balanced is called a balanced dominating set if In this dominating set if paper, we investigate the balanced domination number for some special graphs. Keywords: balanced, domination, The special graph balanced domination number Mathematics subject classification: 05C69 is the minimum cardinality of the balanced weak balanced dominating set dominating set.
Graph Theory Graph Theory (III)

J.A. Bondy U.S.R. Murty Graph Theory (III) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classiﬁcation (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.
Necessary Condition for Cubic Planar 3-Connected Graph to Be Non-Hamiltonian with Proof of Barnette’S Conjecture

International J.Math. Combin. Vol.3(2014), 70-88 Necessary Condition for Cubic Planar 3-Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture Mushtaq Ahmad Shah (Department of Mathematics, Vivekananda Global University (Formerly VIT Jaipur)) E-mail: shahmushtaq81@gmail.com Abstract: A conjecture of Barnette states that, every three connected cubic bipartite pla- nar graph is Hamiltonian. This problem has remained open since its formulation. This paper has a threefold purpose. The first is to provide survey of literature surrounding the conjec- ture. The second is to give the necessary condition for cubic planar three connected graph to be non-Hamiltonian and finally, we shall prove near about 50 year Barnett’s conjecture. For the proof of different results using to prove the results we illustrate most of the results by using counter examples. Key Words: Cubic graph, hamiltonian cycle, planar graph, bipartite graph, faces, sub- graphs, degree of graph. AMS(2010): 05C25 §1. Introduction It is not an easy task to prove the Barnette’s conjecture by direct method because it is very difficult process to prove or disprove it by direct method. In this paper, we use alternative method to prove the conjecture. It must be noted that if any one property of the Barnette’s graph is deleted graph is non Hamiltonian. A planar graph is an undirected graph that can be embedded into the Euclidean plane without any crossings. A planar graph is called polyhedral if and only if it is three vertex connected, that is, if there do not exists two vertices the removal of which would disconnect the rest of the graph.
36 DM14 Abstracts

36 DM14 Abstracts IP0 University of Oxford Hot Topic Session: The Existence of Designs Robert.Hough@maths.ox.ac.uk We prove the existence conjecture for combinatorial de- signs, answering a question of Steiner from 1853. More IP4 generally, we show that the natural divisibility conditions Submodular Functions and Their Applications are suﬃcient for clique decompositions of simplicial com- plexes that satisfy a certain pseudorandomness condition. Submodular functions, a discrete analogue of convex func- tions, have played a fundamental role in combinatorial op- timization since the 1970s. In the last decade, there has Peter Keevash been renewed interest in submodular functions due to their University of Oxford role in algorithmic game theory, as well as numerous appli- Peter.Keevash@maths.ox.ac.uk cations in machine learning. These developments have led to new questions as well as new algorithmic techniques. I will discuss the concept of submodularity, its unifying role IP1 in combinatorial optimization, and a few illustrative appli- The Graph Regularity Method cations. I will describe some recent algorithmic advances, in particular the concept of multilinear relaxation, and the Szemerdi’s regularity lemma is one of the most powerful role of symmetry in proving hardness results for submod- tools in graph theory, with many applications in combina- ular optimization. I will conclude by discussing possible torics, number theory, discrete geometry, and theoretical extensions of the notion of submodularity, and some fu- computer science. Roughly speaking, it says that every ture challenges. large graph can be partitioned into a small number of parts such that the bipartite subgraph between almost all pairs Jan Vondrak of parts is random-like.
Graphs, Algorithms, and Optimization Discrete Mathematics and Its Applications Author: Kocay, William.; Kreher, Donald L

cover Cover title: Graphs, Algorithms, and Optimization Discrete Mathematics and Its Applications author: Kocay, William.; Kreher, Donald L. publisher: CRC Press isbn10 | asin: 0203489055 print isbn13: 9780203620892 ebook isbn13: 9780203489055 language: English subject Graph algorithms. publication date: 2005 lcc: QA166.245.K63 2005eb ddc: 511/.5 subject: Graph algorithms. cover Page i DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H.ROSEN GRAPHS, ALGORITHMS, AND OPTIMIZATION page_i Page ii DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor Kenneth H.Rosen, Ph.D. Charles J.Colbourn and Jeffrey H.Dinitz, The CRC Handbook of Combinatorial Designs Charalambos A.Charalambides, Enumerative Combinatorics Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E.Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan Gross and Jay Yellen, Graph Theory and Its Applications Jonathan Gross and Jay Yellen, Handbook of Graph Theory Darrel R.Hankerson, Greg A.Harris, and Peter D.Johnson, Introduction to Information Theory and Data Compression, Second Edition Daryl D.Harms, Miroslav Kraetzl, Charles J.Colbourn, and John S.Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment David M.Jackson and Terry I.Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E.Klima, Ernest Stitzinger, and Neil P.Sigmon, Abstract
Title Characterization of Bipartite Graph and Its Hamiltonicity All

Characterization of Bipartite Graph and its Hamiltonicity Title Shwin Seinn All Authors Local publication Publication Type Publisher Universities Research Journal, Department of Higher Education Vol.I, No.2 (Journal name, issue no., page no etc.) In this paper, the characterization of bipartite graph will be first described by using the definitions of cycle, Hamiltonian cycle, odd cycle, and chromatic number in a graph. Then, the conditions for a bipartite graph to be Hamiltonian are investigated. Furthermore, a new necessary condition will be Abstract shown for the Hamiltonicity of bipartite graph. Bipartite Graph, Hamitonian Cycle, Hamiltinicity Keywords Citation 2008 Issue Date Characterization of Bipartite Graph and Its Hamiltonicity SHWIN SEINN* Abstract In this paper, the characterization of bipartite graph will be first described by using the definitions of cycle, Hamiltonian cycle, odd cycle, and chromatic number in a graph. Then, the conditions for a bipartite graph to be Hamiltonian are investigated. Furthermore, a new necessary condition will be shown for the Hamiltonicity of bipartite graph. Keywords: Bipartite Graph, Hamiltonian cycle, Hamiltonicity Introduction The most useful object in discrete mathematics is a structure called a graph. Graphs were first introduced in the eighteenth century by the Swiss Mathematician Leonard Euler. One of the reasons for the recent interest in graph theory is its applicability in many diverse fields including computer science, chemistry, electrical engineering, and economics. In the study of graph theory, it is necessary to determine the existence of a cycle in which all the vertices appear exactly once except for the starting and ending vertex that appears twice. Such a cycle is called a Hamiltonian cycle and we say a graph is Hamiltonian if it contains a Hamiltonian cycle.