1The BasicsThis chapter gives a gentle yet concise introduction to most of the ter-minology used later in the book. Fortunately, much of standard graphtheoretic terminology is so intuitive that it is easy to remember; the fewterms better understood in their proper setting will be introduced later,when their time has coction 1.1 offers a brief but self-contained sumryof themostbasic definitions in graph theory, those centred round the notion of agraph. Most readers will have met these definitions before, or will havethem explained to them as they begin to read this book. For this reason,Section 1.1 does not dwell on these definitions more than clarity requires:its mainpurpose is to collect themost basicterms in one plaoloreasyreference later. For deviations for multigraphs see Section 1.10.From Section 12onwards.all newdefinitions will bebroughttolifealmostimnmediatelybyanumber ofsimplevetfundamental propositionsOften, these will relate the newly defined terms to one another: theof how the vaof oneriant influences that of anotheTOuunderlies much of graph theory, and it will be good to become familiarwith this line of thinking early.By N we denote the set of natural numbers, including zero. The setZ/nZ of integers modulo n is denoted by Zn; its elements are writenZ,asi:=i+nzWhen weregard Z,=0.i)as afield,wealso denoteit as F, - [0,1]. For a real mumber we denote by [] the greatest[] [a]integer .and by[r the least integer ≥ r.Logarithms written aslog' are taken at baoase2:thenatural logarithmwillbedenotedby"lnlog, InTan that a is being defined aand:E·,Ax) of disjoint subsets of a set A is a partisetA-fAorpartof Aif the union UAof all the sets A, e A is A and A; + O for every i.UAAnother partition [A', .., A] of A refines the partition A if each A, is[4]contained in some Aj. By [A]* we denote the set of all k-element subsetsof A. Sets with k elements will be called k-sets: subsets with k elementare k-subsets.k-set Reinhard Diestel 2017RTTexts1natics173GrapnTnDOI10.1007/978-3-662-53622-3_1 The Basics This chapter gives a gentle yet concise introduction to most of the ter￾minology used later in the book. Fortunately, much of standard graph theoretic terminology is so intuitive that it is easy to remember; the few terms better understood in their proper setting will be introduced later, when their time has come. Section 1.1 offers a brief but self-contained summary of the most basic definitions in graph theory, those centred round the notion of a graph. Most readers will have met these definitions before, or will have them explained to them as they begin to read this book. For this reason, Section 1.1 does not dwell on these definitions more than clarity requires: its main purpose is to collect the most basic terms in one place, for easy reference later. For deviations for multigraphs see Section 1.10. From Section 1.2 onwards, all new definitions will be brought to life almost immediately by a number of simple yet fundamental propositions. Often, these will relate the newly defined terms to one another: the question of how the value of one invariant influences that of another underlies much of graph theory, and it will be good to become familiar with this line of thinking early. By N we denote the set of natural numbers, including zero. The set Z/nZ of integers modulo n is denoted by Zn; its elements are written Zn as i := i + nZ. When we regard Z2 = {0, 1} as a field, we also denote it as F2 = {0, 1}. For a real number x we denote by x the greatest integer x, and by x the least integer  x. Logarithms written as x, x ‘log’ are taken at base 2; the natural logarithm will be denoted by ‘ln’. log, ln The expressions x := y and y =: x mean that x is being defined as y. A set A = {A1,...,Ak} of disjoint subsets of a set A is a partition partition of A if the union A of all the sets Ai ∈ A is A and Ai = ∅ for every i. A Another partition {A 1,...,A } of A refines the partition A if each A i is contained in some Aj . By [A] k we denote the set of all k-element subsets [A] k of A. Sets with k elements will be called k-sets; subsets with k elements are k-subsets. k-set R. Diestel, Graph Theory, Graduate Texts in Mathematics 173, DOI 10.1007/978-3-662-53622-3_1 © Reinhard Diestel 2017 1