Not Every Bipartite Double Cover Is Canonical
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Automorphisms of the Double Cover of a Circulant Graph of Valency at Most 7
AUTOMORPHISMS OF THE DOUBLE COVER OF A CIRCULANT GRAPH OF VALENCY AT MOST 7 ADEMIR HUJDUROVIC,´ ¯DOR¯DE MITROVIC,´ AND DAVE WITTE MORRIS Abstract. A graph X is said to be unstable if the direct product X × K2 (also called the canonical double cover of X) has automorphisms that do not come from automorphisms of its factors X and K2. It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors. We find all of the nontrivially unstable circulant graphs of valency at most 7. (They come in several infinite families.) We also show that the instability of each of these graphs is explained by theorems of Steve Wilson. This is best possible, because there is a nontrivially unstable circulant graph of valency 8 that does not satisfy the hypotheses of any of Wilson's four instability theorems for circulant graphs. 1. Introduction Let X be a circulant graph. (All graphs in this paper are finite, simple, and undirected.) Definition 1.1 ([16]). The canonical bipartite double cover of X is the bipartite graph BX with V (BX) = V (X) 0; 1 , where × f g (v; 0) is adjacent to (w; 1) in BX v is adjacent to w in X: () Letting S2 be the symmetric group on the 2-element set 0; 1 , it is clear f g that the direct product Aut X S2 is a subgroup of Aut BX. We are interested in cases where this subgroup× is proper: Definition 1.2 ([12, p. -
Graph Operations and Upper Bounds on Graph Homomorphism Counts
Graph Operations and Upper Bounds on Graph Homomorphism Counts Luke Sernau March 9, 2017 Abstract We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n-vertex, d-regular graph G and any graph H (possibly with loops), n n d d hom(G, H) ≤ max hom(Kd,d,H) 2 , hom(Kd+1,H) +1 , n o where hom(G, H) is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on hom(G, H) holds for all n-vertex, d-regular G. In particular we show that if HWR is the complete looped path on three vertices, also known as the Widom-Rowlinson graph, then n d hom(G, HWR) ≤ hom(Kd+1,HWR) +1 for all n-vertex, d-regular G. This verifies a conjecture of Galvin. arXiv:1510.01833v3 [math.CO] 8 Mar 2017 1 Introduction Graph homomorphisms are an important concept in many areas of graph theory. A graph homomorphism is simply an adjacency-preserving map be- tween a graph G and a graph H. That is, for given graphs G and H, a function φ : V (G) → V (H) is said to be a homomorphism from G to H if for every edge uv ∈ E(G), we have φ(u)φ(v) ∈ E(H) (Here, as throughout, all graphs are simple, meaning without multi-edges, but they are permitted 1 to have loops). -
Coverings of Cubic Graphs and 3-Edge Colorability
Discussiones Mathematicae Graph Theory 41 (2021) 311–334 doi:10.7151/dmgt.2194 COVERINGS OF CUBIC GRAPHS AND 3-EDGE COLORABILITY Leonid Plachta AGH University of Science and Technology Krak´ow, Poland e-mail: [email protected] Abstract Let h: G˜ → G be a finite covering of 2-connected cubic (multi)graphs where G is 3-edge uncolorable. In this paper, we describe conditions under which G˜ is 3-edge uncolorable. As particular cases, we have constructed regular and irregular 5-fold coverings f : G˜ → G of uncolorable cyclically 4-edge connected cubic graphs and an irregular 5-fold covering g : H˜ → H of uncolorable cyclically 6-edge connected cubic graphs. In [13], Steffen introduced the resistance of a subcubic graph, a char- acteristic that measures how far is this graph from being 3-edge colorable. In this paper, we also study the relation between the resistance of the base cubic graph and the covering cubic graph. Keywords: uncolorable cubic graph, covering of graphs, voltage permuta- tion graph, resistance, nowhere-zero 4-flow. 2010 Mathematics Subject Classification: 05C15, 05C10. 1. Introduction 1.1. Motivation and statement of results We will consider proper edge colorings of G. Let ∆(G) be the maximum degree of vertices in a graph G. Denote by χ′(G) the minimum number of colors needed for (proper) edge coloring of G and call it the chromatic index of G. Recall that according to Vizing’s theorem (see [14]), either χ′(G) = ∆(G), or χ′(G) = ∆(G)+1. In this paper, we shall consider only subcubic graphs i.e., graphs G such that ∆(G) ≤ 3. -
Lombardi Drawings of Graphs 1 Introduction
Lombardi Drawings of Graphs Christian A. Duncan1, David Eppstein2, Michael T. Goodrich2, Stephen G. Kobourov3, and Martin Nollenburg¨ 2 1Department of Computer Science, Louisiana Tech. Univ., Ruston, Louisiana, USA 2Department of Computer Science, University of California, Irvine, California, USA 3Department of Computer Science, University of Arizona, Tucson, Arizona, USA Abstract. We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. 1 Introduction The American artist Mark Lombardi [24] was famous for his drawings of social net- works representing conspiracy theories. Lombardi used curved arcs to represent edges, leading to a strong aesthetic quality and high readability. Inspired by this work, we intro- duce the notion of a Lombardi drawing of a graph, in which edges are drawn as circular arcs with perfect angular resolution: consecutive edges are evenly spaced around each vertex. While not all vertices have perfect angular resolution in Lombardi’s work, the even spacing of edges around vertices is clearly one of his aesthetic criteria; see Fig. 1. Traditional graph drawing methods rarely guarantee perfect angular resolution, but poor edge distribution can nevertheless lead to unreadable drawings. Additionally, while some tools provide options to draw edges as curves, most rely on straight-line edges, and it is known that maintaining good angular resolution can result in exponential draw- ing area for straight-line drawings of planar graphs [17,25]. -
The Operations Invariant Properties on Graphs Yanzhong Hu, Gang
International Conference on Education Technology and Information System (ICETIS 2013) The Operations Invariant Properties on Graphs Yanzhong Hu, Gang Cheng School of Computer Science, Hubei University of Technology, Wuhan, 430068, China [email protected], [email protected] Keywords:Invariant property; Hamilton cycle; Cartesian product; Tensor product Abstract. To determine whether or not a given graph has a Hamilton cycle (or is a planar graph), defined the operations invariant properties on graphs, and discussed the various forms of the invariant properties under the circumstance of Cartesian product graph operation and Tensor product graph operation. The main conclusions include: The Hamiltonicity of graph is invariant concerning the Cartesian product, and the non-planarity of the graph is invariant concerning the tensor product. Therefore, when we applied these principles into practice, we testified that Hamilton cycle does exist in hypercube and the Desargues graph is a non-planarity graph. INTRODUCTION The Planarity, Eulerian feature, Hamiltonicity, bipartite property, spectrum, and so on, is often overlooked for a given graph. The traditional research method adopts the idea of direct proof. In recent years, indirect proof method is put forward, (Fang Xie & Yanzhong Hu, 2010) proves non- planarity of the Petersen graph, by use the non-planarity of K3,3, i.e., it induces the properties of the graph G with the properties of the graph H, here H is a sub-graph of G. However, when it comes to induce the other properties, the result is contrary to what we expect. For example, Fig. 1(a) is a Hamiltonian graph which is a sub-graph of Fig. -
Nearly Tight Approximability Results for Minimum Biclique Cover and Partition
Nearly Tight Approximability Results for Minimum Biclique Cover and Partition Parinya Chalermsook1, Sandy Heydrich1;3, Eugenia Holm2, and Andreas Karrenbauer1∗ 1 Max Planck Institute for Informatics, Saarbr¨ucken, Germany 2 Dept. for Computer and Information Science, University of Konstanz, Germany 3 Saarbr¨ucken Graduate School of Computer Science {andreas.karrenbauer,parinya.chalermsook,sandy.heydrich}@mpi-inf.mpg.de [email protected] Abstract. In this paper, we consider the minimum biclique cover and minimum biclique partition problems on bipartite graphs. In the min- imum biclique cover problem, we are given an input bipartite graph G = (V; E), and our goal is to compute the minimum number of complete bipartite subgraphs that cover all edges of G. This problem, besides its correspondence to a well-studied notion of bipartite dimension in graph theory, has applications in many other research areas such as artificial intelligence, computer security, automata theory, and biology. Since it is NP-hard, past research has focused on approximation algorithms, fixed parameter tractability, and special graph classes that admit polynomial time exact algorithms. For the minimum biclique partition problem, we are interested in a biclique cover that covers each edge exactly once. We revisit the problems from approximation algorithms' perspectives and give nearly tight lower and upper bound results. We first show that both problems are NP-hard to approximate to within a factor of n1−" (where n is the number of vertices in the input graph). Using a stronger complexity assumption, the hardness becomes Ω~(n), where Ω~(·) hides lower order terms. Then we show that approximation factors of the form n=(log n)γ for some γ > 0 can be obtained. -
Enlarging Properties of Graphs
Enlarging Properties of Graphs Alan Geoffrey Boshier PhD Thesis, Royal Holloway & Bedford New College, University of London. gv^A y COLLEGE UBRARY O ProQuest Number: 10096264 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10096264 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Abstract The subjects of this Thesis are the enlarging and magnifying properties of graphs. Upper bounds for the isoperimetric number i{G) of a graph G are determined with respect to such elementary graph properties as order, valency, and the number of three and four- cycles. The relationship between z(G) and the genus of G is studied in detail, and a class of graphs called finite element graphs is shown never to supply enlarging families. The magnifying properties of Hamiltonian cubic graphs are investigated, and a class of graphs known as shift graphs is defined. These are shown never to form enlarging families, using a technical lemma derived from Klawe’s Theorem on non-expanding families of graphs. The same lemma is used, in conjunction with some elementary character theory, to prove that several important classes of Cayley graphs do not form enlarging families, and to derive a lower bound on the subdominant eigenvalue of a vertex-transitive graph. -
Free Groups and Graphs: the Hanna Neumann Theorem
FREE GROUPS AND GRAPHS: THE HANNA NEUMANN THEOREM LAUREN COTE Abstract. The relationship between free groups and graphs is one of the many examples of the interaction between seemingly foreign branches of math- ematics. After defining and proving elementary relations between graphs and groups, this paper will present a result of Hanna Neumann on the intersection of free subgroups. Although Neumann's orginal upper bound on the rank of the intersection has been improved upon, her conjecture, that the upper bound is half of the one she originally proved, remains an open question. Contents 1. Groups 1 2. Graphs 3 3. Hanna Neumann Theorem 9 References 11 1. Groups Definition 1.1. Recall that a group G is a set S and a binary operation · with the properties that G is closed under ·, · is associative, there exists an identity e such that for all g 2 (G); g · e = e · g = g, and there exist inverses such that for each g 2 G; there exists g−1 2 G such that g · g−1 = g−1 · g = e. When working with groups, it is helpful to review the definitions of homomor- phism, an operation perserving map. For groups G1 = (S1; ·), G2 = (S2;?), the homomorphism φ : G1 ! G2 has the property that φ(x · y) = φ(x) ? φ(y). An isomorphism is a bijective homomorphism. Definition 1.2. This paper will focus on a particular kind of group, those which have the least possible amount of restriction: free groups. Let i : S ! F . A group F is free on a set S if for all groups G and for all f : S ! G, there exists a unique homomorphism : F ! G so that i ◦ = f. -
Enumerating Branched Orientable Surface Coverings Over a Non-Orientable Surface Ian P
Discrete Mathematics 303 (2005) 42–55 www.elsevier.com/locate/disc Enumerating branched orientable surface coverings over a non-orientable surface Ian P. Gouldena, Jin Ho Kwakb,1, Jaeun Leec,2 aDepartment of Combinatorics and Optimization, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 bCombinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang 790–784, Korea cMathematics, Yeungnam University, Kyongsan 712–749, Korea Received 21 November 2002; received in revised form 13 October 2003; accepted 29 October 2003 Available online 10 October 2005 Abstract The isomorphism classes ofseveral types ofgraph coverings ofa graph have been enumerated by many authors [M. Hofmeister, Graph covering projections arising from finite vector space over finite fields, Discrete Math. 143 (1995) 87–97; S. Hong, J.H. Kwak, J. Lee, Regular graph cover- ings whose covering transformation groups have the isomorphism extention property, Discrete Math. 148 (1996) 85–105; J.H. Kwak, J.H. Chun, J.Lee, Enumeration ofregular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273–285; J.H. Kwak, J. Lee, Isomorphism classes ofgraph bundles, Canad. J. Math. XLII (1990) 747–761; J.H. Kwak, J. Lee, Enumeration ofconnected graph coverings, J. Graph Theory 23 (1996) 105–109]. Recently, Kwak et al [Balanced regular coverings ofa signed graph and regular branched orientable surface coverings over a non-orientable surface, Discrete Math. 275 (2004) 177–193] enumerated the isomorphism classes ofbalanced regular coverings ofa signed graph, as a continuation ofan enumeration work done by Archdeacon et al [Bipartite covering graphs, Discrete Math. 214 (2000) 51–63] the isomorphism classes ofbranched orientable regular surfacecoverings ofa non-orientable surface having a finite abelian covering transformation group. -
Bidirected Graph from Wikipedia, the Free Encyclopedia Contents
Bidirected graph From Wikipedia, the free encyclopedia Contents 1 Bidirected graph 1 1.1 Other meanings ............................................ 1 1.2 See also ................................................ 2 1.3 References ............................................... 2 2 Bipartite double cover 3 2.1 Construction .............................................. 3 2.2 Examples ............................................... 3 2.3 Matrix interpretation ......................................... 4 2.4 Properties ............................................... 4 2.5 Other double covers .......................................... 4 2.6 See also ................................................ 5 2.7 Notes ................................................. 5 2.8 References ............................................... 5 2.9 External links ............................................. 6 3 Complex question 7 3.1 Implication by question ........................................ 7 3.2 Complex question fallacy ....................................... 7 3.2.1 Similar questions and fallacies ................................ 8 3.3 Notes ................................................. 8 4 Directed graph 10 4.1 Basic terminology ........................................... 11 4.2 Indegree and outdegree ........................................ 11 4.3 Degree sequence ............................................ 12 4.4 Digraph connectivity .......................................... 12 4.5 Classes of digraphs ......................................... -
Totally Silver Graphs
Totally Silver Graphs M. Ghebleha, E. S. Mahmoodianb aDepartment of Mathematics, Faculty of Science, Kuwait University, State of Kuwait bDepartment of Mathematical Sciences, Sharif University of Technology, Tehran, Iran Abstract A totally silver coloring of a graph G is a k{coloring of G such that for every vertex v 2 V (G), each color appears exactly once on N[v], the closed neighborhood of v.A totally silver graph is a graph which admits a totally silver coloring. Totally silver coloring are directly related to other areas of graph theory such as distance coloring and domination. In this work, we present several constructive characterizations of totally silver graphs and bipartite totally silver graphs. We give several infinite families of totally silver graphs. We also give cubic totally silver graphs of girth up to 10. Keywords: Graph coloring, distance coloring, silver coloring, domination. 1. Introduction All graphs in this paper are finite and simple. A totally silver coloring of a graph G is a k{coloring of G such that for every vertex v 2 V (G), each color appears exactly once on N[v], the closed neighborhood of v.A totally silver graph is a graph which admits a totally silver coloring. It is immediate from this definition that every graph admitting a totally silver coloring with k colors is (k − 1){regular. Totally silver colorings are introduced in [4]. The term totally silver is inspired by an International Mathematical Olympiad problem on silver matrices [6]. Totally silver colorings appear, under the name perfect colorings, in the study of file seg- mentations in a network in [1]. -
An Analysis of the Properties of Various Interconnection Networks K
Volume 5, No. 4, April 2014 (Special Issue) ISSN No. 0976-5697 International Journal of Advanced Research in Computer Science REVIEW ARTICAL Available Online at www.ijarcs.info An Analysis of the Properties of Various Interconnection Networks K. Karthik Sudarson Jena Dept .of CSE Dept. of IT BVRIT, Narsapur GITAM University [email protected] [email protected] T. Venu Gopal Dept. of CSE JNTUHCEJ, Kondagattu [email protected] Abstract: In this paper, different types of interconnection networks are investigated and some of their properties are analyzed to summarize the differences in their network cost. The various properties of the interconnection networks such as connectivity, routing algorithm, diameter and broadcasting technology are investigated. This analysis gives a framework for the construction of more efficient interconnection networks in future. Key words: Interconnection Networks, Network cost, Routing Algorithms, Diameter, Broad casting Technology Network scales being typically used for comparative I. INTRODUCTION evaluation of an interconnection network due to the said characteristic include network cost [4-10] defined as degree x In parallel computing, to achieve parallelism various diameter of an interconnection network. By virtue of its merit techniques can be employed. One technique of achieving of easily providing a communication network system required parallelism is using MIMD (Multiple Instruction Multiple in applications of all kinds. Hypercube is node-symmetric and Data stream). Machines using MIMD have a number of edge-symmetric, has a simple routing algorithm with maximal processor that functions asynchronously and independently. fault tolerance and a simple reflexive system, and also has a At any time, different processors may be executing different merit that it may be readily embedded with the proposed instructions on different pieces of data.