DOCSLIB.ORG

Topics in Trivalent Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topics in Trivalent Graphs Topics in Trivalent Graphs Marijke van Gans January sssssssssssssssssssss sssssssssss ss s sss ssssssssss ssssss ss. .......s. .. .s .s sssssss ssssssss .s .... .. ........s.......................... ssssssss ssss ................ .s.. ...s ......... ssss ssss ........ ......... sss sssss ..... ..... ssss ssss.s .... ..... sssss sssss... ....s..... sssss ssss ..... .ssss sss.s . ....sssss ssss.................s .. .... sss ss ... .. .....s.. sss sss .. .. .. ssss ssss .. .. ......s........ sss ssss..s ..s .. ss sss ........... ........ .. .ssss sss ....s ......... ......... s. ...s ........s. ssss ssss .......... .... .......... .... sss ss ........s.......... .. .. ss ssss ...... .. ss ss......s.... .. sss ss....s... ..ssss ss..s.. .. ..s... s........ s sss.s. .... ................. sss ss .. ... .........s.. ss sss . ... .....sss s .. .. .sss s . .. ss s .. .. s s s..................s. s s . .. s s .. .. sss ss . ... sss sss .. .. .s....ss ss . ... .........ssss ss.ss .. ... ..........s.. s ss....s . ...s.......... ..... ssss sss...... .....sss sss.....s. .. .sss ss.. .....s . ss ssss............. .. .. ss ss.s.... .......s s.. ..s ssss ss ............. ......... ..................s..... ss ssss s.... ...... s........ .. ..sss ss......... ... .. sss ss..s .. .. ...s ssss sss .. .. ............sss ss ... .. .sssss sssss ...s. .. ..s.....sss ss.s...s............. ..s...sss ssss .s .. ..s..ssss sss......s ..s.. sssss ss..sss .... ..... sssss ssss ..... ..... ssss ssss .s... ....s sss ssssssss s............ ............. ssss ss.s..s....s...s..s ..s ... ........s....... s s sssssssssss ssssss.s....sss ..s ..s ....s.........s.....s..s... sssss ssssssssssss..s.s..s...sssssssssss.sssss.sssss A thesis submitted to The University of Birmingham for the degree of Doctor of Philosophy University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract Chapter 0 details the notation and terminology used. Chapter 1 introduces the usual linear algebra over IF 2 of edge space E and its orthogonal subspaces Z (cycle space) and Z⊥ (cut space). Reduced vectors are defined as elements of the quotient space E/Z⊥. Reduced vectors of edges give a simple way of characterising edges that are bridges (their reduced vector is null) or 2-edge cuts (their vectors are equal), and also of spanning trees (the edges outside the tree are a basis) and form to the best of my knowledge a new approach. They are also useful in later chapters to describe Tait colorings, as well as cycle double covers. Perhaps the most important property of E/Z⊥ is the Unique graph theorem: unlike in E, a list of which reduced vectors are edges uniquely determines graph structure (if edge connectivity is high enough; that covers certain “solid” components every trivalent graph can be decomposed into). Chapter 2 gives a brief intoduction to graph embeddings and planar graphs. Chapter 3 deals specifically with trivalent graphs, listing some of the ways in which they are different from graphs in general. Results here include two versions of Bipolar growth theorem which can be used for constructive proofs, and (after Qs defining “halftrees” and a “flipping” operation Jss ↔ s between them) a theorem J QQ enumerating the set Cn of halftrees of a given size, the Caterpillar theorem showing Cn is connected by flipping, and the Butterfly theorem derived from it. Graphs referred to here as solid are shown to play an important structural rˆole. Chapter 4 deals with the 4-coloring theorem. The first half shows the older results in a unified light using edge spaces over IF 4. The second half applies methods from coding theory to this. The 4-color theorem is shown to be equivalent to a variety of statements about cycle-shaped words in codes over IF 4 or IF 3,manyofthem tantalisingly simple to state (but not, as yet, to prove). Chapter 5 deals with what has been variously called polyhedral decompositions and (specifically for those using cycles) cycle double covers, as in the cycle dou- ble cover conjecture. The more general concept is referred to as a map in this paper, and identified with what is termed here cisness structures,whichisa new approach. There is also a simpler proof of a theorem by Szekeres. Links with the subject of the previous chapter are identified, and some approaches towards proving the conjecture suggested. Several planned appendices were left out of the version submitted for examination because they would make the thesis too big, and/or were not finished. Of the ones that remain, appendix H (on embedding infinite 4- and 3-valent trees X∞ and Y∞ in the hyperbolic plane) now seems disjointed from the body of the text (a planned appendix dealt with colorings of finite graphs as the images of homomorphisms from embeddings of Y∞). Appendix B enumerates cycle maps (cycle double covers) on a number of small graphs while appendix D investigates dim Z ∩ Z⊥. 2 Concrete Nodes are in small caps, edges (and halfedges) in lower case γρεεκ. Integers and field elements are in lower case italic,mostly. Spaces (linear codes) over IF 2 are in UPPER and their vectors in lower case bold. Spaces (linear codes) over IF 4 are in UPPER and their vectors in lower case bold italic; sets (codes) that aren’t subspaces are written like THIS. n nodes, m edges, k components; o := m − n + k; if trivalent n/2=:h := m/3. We is the mathematical community at large when the subject of verbs such as “call”; it is the readers and author together in phrases like “. will come across an example of this in the next section”. I is used when referring to me. a This paper was written in LTEX with a sprinkling of plain TEX. a draw The contents of the LTEX picture environments were designed with pict+ . Thanks • to Rob T. Curtis and the gang for putting up with me all this time. • to the examiners for their helpful corrections and suggestions. • to Rudy J. List and Bernard B. Beard for believing in me. • to Bill Daly for rekindling my interest in trivalent graphs. • and everybody else at CIS:SCIMATH for all the great stuff, 1994–2004. • to h for, among other things, feeding me when other funding dried up. 3 Contents 0 Graphs 6 0.0 Terminology ............................... 6 0.1 Connectivity ............................... 7 0.2 Cyclicity ................................. 9 1 Spaces 12 1.0 Edge space E .............................. 12 1.1 Cut space Z⊥ .............................. 12 1.2 Cycle space Z .............................. 13 1.3 Orthogonality .............................. 14 1.4 Halfgraphs ................................ 16 1.5 Reduced edge space E◦ ......................... 16 1.6 E◦ is dual to Z ............................. 19 2Planarity 21 2.0 Embedding in surfaces ......................... 21 2.1 Planar graphs .............................. 22 3Trivalence 24 3.0 All the graphs you’ll ever need? .................... 24 3.1 Growing trivalent graphs ........................ 25 3.2 Spanning trees ............................. 29 3.3 Flips of trivalent graphs ........................ 30 3.4 All planar trivalent halftrees ...................... 32 3.5 All planar trivalent graphs ....................... 36 3.6 All trivalent graphs ........................... 39 4 Tait colorings 41 4.0 Colorings ................................ 41 4.1 Planar face-4-coloring .......................... 42 4.2 Tait’s edge-3-coloring .......................... 45 4.3 Heawood’s node-2-coloring ....................... 47 4.4 The elusive 1-coloring .......................... 49 4.5 Spaces and codes for Tait colorings .................. 51 4.6 Variations on a theme ......................... 53 4.6.0 Taking stock ........................... 53 4.6.1 The constructive view ..................... 55 4.6.2 Equatorial codes ........................ 56 4.6.3 Additive description ...................... 57 4.6.4 Multiplicative description ................... 61 4.6.5 log Heawood color space .................... 65 4.6.6 Quark confinement ....................... 66 4 4.6.7 Weights and the dual code ................... 69 4.6.8 Complete weights ........................ 71 4.6.9 Tait coloring reduced vectors .................. 72 5 Cycle double covers 75 5.0 Maps ................................... 75 5.1 Oriented maps .............................. 78 5.2 Cisness .................................. 81 5.3 Cycle maps ............................... 84 5.4 Combinatorial genus .......................... 86 5.5 Combining maps ............................ 88 5.6 Existence lemmata ........................... 90 5.7 Path counts ............................... 93 5.8 Mapping the graph ........................... 95 Index 100 References 103 Appendices B, D and H are in separate files (with their own lists of references). 5 0 Graphs 0.0 Terminology Graph theory nomenclature being notoriously variable, it is perhaps best to briefly define the terms as used in this paper. A graph (formerly simple graph) consists of a set N of items called nodes (vertices, points etc.), with a set E⊆the set of (unordered) pairs of nodes. The pairs that are
Load more

Recommended publications
  • Directed Cycle Double Cover Conjecture: Fork Graphs
    Directed Cycle Double Cover Conjecture: Fork Graphs Andrea Jim´enez ∗ Martin Loebl y October 22, 2013 Abstract We explore the well-known Jaeger's directed cycle double cover conjecture which is equiva- lent to the assertion that every cubic bridgeless graph has an embedding on a closed orientable surface with no dual loop. We associate each cubic graph G with a novel object H that we call a hexagon graph; perfect matchings of H describe all embeddings of G on closed orientable surfaces. The study of hexagon graphs leads us to define a new class of graphs that we call lean fork-graphs. Fork graphs are cubic bridgeless graphs obtained from a triangle by sequentially connecting fork-type graphs and performing Y−∆, ∆−Y transformations; lean fork-graphs are fork graphs fulfilling a connectivity property. We prove that Jaeger's conjecture holds for the class of lean fork-graphs. The class of lean fork-graphs is rich; namely, for each cubic bridgeless graph G there is a lean fork-graph containing a subdivision of G as an induced subgraph. Our results establish for the first time, to the best of our knowledge, the validity of Jaeger's conjecture in a broad inductively defined class of graphs. 1 Introduction One of the most challenging open problems in graph theory is the cycle double cover conjecture which was independently posed by Szekeres [14] and Seymour [13] in the seventies. It states that every bridgeless graph has a cycle double cover, that is, a system C of cycles such that each edge of the graph belongs to exactly two cycles of C.
    [Show full text]
  • Math 7410 Graph Theory
    Math 7410 Graph Theory Bogdan Oporowski Department of Mathematics Louisiana State University April 14, 2021 Definition of a graph Definition 1.1 A graph G is a triple (V,E, I) where ◮ V (or V (G)) is a finite set whose elements are called vertices; ◮ E (or E(G)) is a finite set disjoint from V whose elements are called edges; and ◮ I, called the incidence relation, is a subset of V E in which each edge is × in relation with exactly one or two vertices. v2 e1 v1 Example 1.2 ◮ V = v1, v2, v3, v4 e5 { } e2 e4 ◮ E = e1,e2,e3,e4,e5,e6,e7 { } e6 ◮ I = (v1,e1), (v1,e4), (v1,e5), (v1,e6), { v4 (v2,e1), (v2,e2), (v3,e2), (v3,e3), (v3,e5), e7 v3 e3 (v3,e6), (v4,e3), (v4,e4), (v4,e7) } Simple graphs Definition 1.3 ◮ Edges incident with just one vertex are loops. ◮ Edges incident with the same pair of vertices are parallel. ◮ Graphs with no parallel edges and no loops are called simple. v2 e1 v1 e5 e2 e4 e6 v4 e7 v3 e3 Edges of a simple graph can be described as v e1 v 2 1 two-element subsets of the vertex set. Example 1.4 e5 e2 e4 E = v1, v2 , v2, v3 , v3, v4 , e6 {{ } { } { } v1, v4 , v1, v3 . v4 { } { }} v e7 3 e3 Note 1.5 Graph Terminology Definition 1.6 ◮ The graph G is empty if V = , and is trivial if E = . ∅ ∅ ◮ The cardinality of the vertex-set of a graph G is called the order of G and denoted G .
    [Show full text]
  • On the Cycle Double Cover Problem
    On The Cycle Double Cover Problem Ali Ghassâb1 Dedicated to Prof. E.S. Mahmoodian Abstract In this paper, for each graph , a free edge set is defined. To study the existence of cycle double cover, the naïve cycle double cover of have been defined and studied. In the main theorem, the paper, based on the Kuratowski minor properties, presents a condition to guarantee the existence of a naïve cycle double cover for couple . As a result, the cycle double cover conjecture has been concluded. Moreover, Goddyn’s conjecture - asserting if is a cycle in bridgeless graph , there is a cycle double cover of containing - will have been proved. 1 Ph.D. student at Sharif University of Technology e-mail: [email protected] Faculty of Math, Sharif University of Technology, Tehran, Iran 1 Cycle Double Cover: History, Trends, Advantages A cycle double cover of a graph is a collection of its cycles covering each edge of the graph exactly twice. G. Szekeres in 1973 and, independently, P. Seymour in 1979 conjectured: Conjecture (cycle double cover). Every bridgeless graph has a cycle double cover. Yielded next data are just a glimpse review of the history, trend, and advantages of the research. There are three extremely helpful references: F. Jaeger’s survey article as the oldest one, and M. Chan’s survey article as the newest one. Moreover, C.Q. Zhang’s book as a complete reference illustrating the relative problems and rather new researches on the conjecture. A number of attacks, to prove the conjecture, have been happened. Some of them have built new approaches and trends to study.
    [Show full text]
  • Closed 2-Cell Embeddings of Graphs with No V8-Minors
    Discrete Mathematics 230 (2001) 207–213 www.elsevier.com/locate/disc Closed 2-cell embeddings of graphs with no V8-minors Neil Robertsona;∗;1, Xiaoya Zhab;2 aDepartment of Mathematics, Ohio State University, Columbus, OH 43210, USA bDepartment of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USA Received 12 July 1996; revised 30 June 1997; accepted 14 October 1999 Abstract A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the Mobius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover. The proof uses a classiÿcation of internally-4-connected graphs with no V8-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Embedding; Strong embedding conjecture; V8 minor; Cycle double cover; Closed 2-cell embedding 1. Introduction The closed 2-cell embedding (called strong embedding in [2,4], and circular em- bedding in [6]) conjecture says that every 2-connected graph G has a closed 2-cell embedding in some surface, that is, an embedding in a compact closed 2-manifold in which each face is simply connected and the boundary of each face is a cycle in the graph (no repeated vertices or edges on the face boundary).
    [Show full text]
  • Algebraically Grid-Like Graphs Have Large Tree-Width
    Algebraically grid-like graphs have large tree-width Daniel Weißauer Department of Mathematics University of Hamburg Abstract By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of ’grid-likeness’ of a graph. The grid contains a long cycle on the perimeter, which is the F2-sum of the rectangles inside. Moreover, the grid distorts the metric of the cycle only by a factor of two. We prove that every graph that resembles the grid in this algebraic sense has large tree-width: Let k,p be integers, γ a real number and G a graph. Suppose that G contains a cycle of length at least 2γpk which is the F2-sum of cycles of length at most p and whose metric is distorted by a factor of at most γ. Then G has tree-width at least k. 1 Introduction For a positive integer n, the (n × n)-grid is the graph Gn whose vertices are all pairs (i, j) with 1 ≤ i, j ≤ n, where two points are adjacent when they are at Euclidean distance 1. The cycle Cn, which bounds the outer face in the natural drawing of Gn in the plane, has length 4(n−1) and is the F2-sum of the rectangles bounding the inner faces. This is by itself not a distinctive feature of graphs with large tree-width: The situation is similar for the n-wheel Wn, arXiv:1802.05158v1 [math.CO] 14 Feb 2018 the graph consisting of a cycle Dn of length n and a vertex x∈ / Dn which is adjacent to every vertex of Dn.
    [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 Flexible Labelings
    Graphs with Flexible Labelings Georg Grasegger∗ Jan Legersk´yy Josef Schichoy For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points con- stant. We give a combinatorial characterization of graphs that have flexible labelings, possibly non-generic. The characterization is based on colorings of the edges with restrictions on the cycles. Furthermore, we give necessary criteria and sufficient ones for the existence of such colorings. 1 Introduction Given a graph together with a labeling of its edges by positive real numbers, we are interested in the set of all functions from the set of vertices to the real plane such that the distance between any two connected vertices is equal to the label of the edge connecting them. Apparently, any such \realization" gives rise to infinitely many equivalent ones, by the action of the group of Euclidean congruence transformations. If the set of equivalence classes is finite and non-empty, then we say that the labeled graph is rigid; if this set is infinite, then we say that the labeling is flexible. The main result in this paper is a combinatorial characterization of graphs that have a flexible labeling. The characterization of graphs such that a generically chosen assign- ment of the vertices to the real plane gives a flexible labeling is classical: by a theorem of Geiringer-Pollaczek [8] which was rediscovered by Laman [6], this is true if the graph contains no Laman subgraph with the same set of vertices. Here, a graph G = (VG;EG) is called a Laman graph if and only if jEGj = 2jVGj − 3 and jEH j ≤ 2jVH j − 3 for any ∗Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences arXiv:1708.05298v2 [math.CO] 19 Nov 2018 yResearch Institute for Symbolic Computation (RISC), Johannes Kepler University Linz This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 675789.
    [Show full text]
  • Cubic Vertex-Transitive Graphs of Girth Six
    CUBIC VERTEX-TRANSITIVE GRAPHS OF GIRTH SIX PRIMOZˇ POTOCNIKˇ AND JANOSˇ VIDALI Abstract. In this paper, a complete classification of finite simple cubic vertex-transitive graphs of girth 6 is obtained. It is proved that every such graph, with the exception of the Desargues graph on 20 vertices, is either a skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of an arc-transitive triangulation of a closed hyperbolic surface, or the truncation of a 6-regular graph with respect to an arc- transitive dihedral scheme. Cubic vertex-transitive graphs of girth larger than 6 are also discussed. 1. Introduction Cubic vertex-transitive graph are one of the oldest themes in algebraic graph theory, appearing already in the classical work of Foster [13, 14] and Tutte [33], and retaining the attention of the community until present times (see, for example, the works of Coxeter, Frucht and Powers [8], Djokovi´cand Miller [9], Lorimer [23], Conder and Lorimer [6], Glover and Maruˇsiˇc[15], Potoˇcnik, Spiga and Verret [27], Hua and Feng [16], Spiga [30], to name a few of the most influential papers). The girth (the length of a shortest cycle) is an important invariant of a graph which appears in many well-known graph theoretical problems, results and formulas. In many cases, requiring the graph to have small girth severely restricts the structure of the graph. Such a phenomenon can be observed when one focuses to a family of graphs of small valence possessing a high level of symmetry. For example, arc-transitive 4-valent graphs of girth at most 4 were characterised in [29].
    [Show full text]
  • Short Cycles
    Short Cycles Minimum Cycle Bases of Graphs from Chemistry and Biochemistry Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium an der Fakultat¨ fur¨ Naturwissenschaften und Mathematik der Universitat¨ Wien Vorgelegt von Petra Manuela Gleiss im September 2001 An dieser Stelle m¨ochte ich mich herzlich bei all jenen bedanken, die zum Entstehen der vorliegenden Arbeit beigetragen haben. Allen voran Peter Stadler, der mich durch seine wissenschaftliche Leitung, sein ub¨ er- w¨altigendes Wissen und seine Geduld unterstutzte,¨ sowie Josef Leydold, ohne den ich so manch tieferen Einblick in die Mathematik nicht gewonnen h¨atte. Ivo Hofacker, dermich oftmals aus den unendlichen Weiten des \Computer Universums" rettete. Meinem Bruder Jurgen¨ Gleiss, fur¨ die Einfuhrung¨ und Hilfstellungen bei meinen Kampf mit C++. Daniela Dorigoni, die die Daten der atmosph¨arischen Netzwerke in den Computer eingeben hat. Allen Kolleginnen und Kollegen vom Institut, fur¨ die Hilfsbereitschaft. Meine Eltern Erika und Franz Gleiss, die mir durch ihre Unterstutzung¨ ein Studium erm¨oglichten. Meiner Oma Maria Fischer, fur¨ den immerw¨ahrenden Glauben an mich. Meinen Schwiegereltern Irmtraud und Gun¨ ther Scharner, fur¨ die oftmalige Betreuung meiner Kinder. Zum Schluss Roland Scharner, Florian und Sarah Gleiss, meinen drei Liebsten, die mich immer wieder aufbauten und in die reale Welt zuruc¨ kfuhrten.¨ Ich wurde teilweise vom osterreic¨ hischem Fonds zur F¨orderung der Wissenschaftlichen Forschung, Proj.No. P14094-MAT finanziell unterstuzt.¨ Zusammenfassung In der Biochemie werden Kreis-Basen nicht nur bei der Betrachtung kleiner einfacher organischer Molekule,¨ sondern auch bei Struktur Untersuchungen hoch komplexer Biomolekule,¨ sowie zur Veranschaulichung chemische Reaktionsnetzwerke herange- zogen. Die kleinste kanonische Menge von Kreisen zur Beschreibung der zyklischen Struk- tur eines ungerichteten Graphen ist die Menge der relevanten Kreis (Vereingungs- menge aller minimaler Kreis-Basen).
    [Show full text]
  • The Distance-Regular Antipodal Covers of Classical Distance-Regular Graphs
    COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 52. COMBINATORICS, EGER (HUNGARY), 198'7 The Distance-regular Antipodal Covers of Classical Distance-regular Graphs J. T. M. VAN BON and A. E. BROUWER 1. Introduction. If r is a graph and "Y is a vertex of r, then let us write r,("Y) for the set of all vertices of r at distance i from"/, and r('Y) = r 1 ('Y)for the set of all neighbours of 'Y in r. We shall also write ry ...., S to denote that -y and S are adjacent, and 'YJ. for the set h} u r b) of "/ and its all neighbours. r i will denote the graph with the same vertices as r, where two vertices are adjacent when they have distance i in r. The graph r is called distance-regular with diameter d and intersection array {bo, ... , bd-li c1, ... , cd} if for any two vertices ry, oat distance i we have lfH1(1')n nr(o)j = b, and jri-ib) n r(o)j = c,(o ::; i ::; d). Clearly bd =co = 0 (and C1 = 1). Also, a distance-regular graph r is regular of degree k = bo, and if we put ai = k- bi - c; then jf;(i) n r(o)I = a, whenever d("f, S) =i. We shall also use the notations k, = lf1b)I (this is independent of the vertex ry),). = a1,µ = c2 . For basic properties of distance-regular graphs, see Biggs [4]. The graph r is called imprimitive when for some I~ {O, 1, ... , d}, I-:/= {O}, If -:/= {O, 1, ..
    [Show full text]
  • Tight Bounds for Online Edge Coloring
    Tight Bounds for Online Edge Coloring Ilan Reuven Cohen∗1, Binghui Peng†‡2, and David Wajc§¶k3 1Carnegie Mellon University and University of Pittsburgh 2Tsinghua University 3Carnegie Mellon University Abstract Vizing’s celebrated theorem asserts that any graph of maximum degree ∆ admits an edge coloring using at most ∆ + 1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2∆ 1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it− only applies to low- degree graphs, with ∆ = O(log n), and they conjectured the existence of online algorithms using ∆(1 + o(1)) colors for ∆= ω(log n). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS’03 and Bahmani et al., SODA’10). We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+ o(1))∆-edge-coloring algorithm for ∆= ω(log n) known a priori. Surprisingly, if ∆ is not known ahead of time, we show that no e Ω(1) ∆-edge-coloring algorithm exists. e−1 − We then provide an optimal, e + o(1) ∆-edge-coloring algorithm for unknown ∆= ω(log n). e−1 Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms. ∗Email address: [email protected]. †Email address: [email protected]. ‡Work done in part while the author was visiting Carnegie Mellon University.
    [Show full text]
  • Hamilton Cycle Heuristics in Hard Graphs (Under the Direction of Pro- Fessor Carla D
    ABSTRACT SHIELDS, IAN BEAUMONT Hamilton Cycle Heuristics in Hard Graphs (Under the direction of Pro- fessor Carla D. Savage) In this thesis, we use computer methods to investigate Hamilton cycles and paths in several families of graphs where general results are incomplete, including Kneser graphs, cubic Cayley graphs and the middle two levels graph. We describe a novel heuristic which has proven useful in finding Hamilton cycles in these families and compare its performance to that of other algorithms and heuristics. We describe methods for handling very large graphs on personal computers. We also explore issues in reducing the possible number of generating sets for cubic Cayley graphs generated by three involutions. Hamilton Cycle Heuristics in Hard Graphs by Ian Beaumont Shields A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Computer Science Raleigh 2004 APPROVED BY: ii Dedication To my wife Pat, and my children, Catherine, Brendan and Michael, for their understanding during these years of study. To the memory of Hanna Neumann who introduced me to combinatorial group theory. iii Biography Ian Shields was born on March 10, 1949 in Melbourne, Australia. He grew up and attended school near the foot of the Dandenong ranges. After a short time in actuarial work he attended the Australian National University in Canberra, where he graduated in 1973 with first class honours in pure mathematics. Ian worked for IBM and other companies in Australia, Canada and the United States, before resuming part-time study for a Masters Degree in Computer Science at North Carolina State University which he received in May 1995.
    [Show full text]