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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of Birmingham for the degree of Doctor of Philosophy University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties.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
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