MATH0029 Graph Theory and Combinatorics

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MATH0029 Graph Theory and Combinatorics Year: 2021{2022 Code: MATH0029 Level: 6 (UG)/7(PG) Normal student group(s): UG Year 3 Mathematics degrees Value: 15 credits (= 7.5 ECTS credits) Term: 1 Assessment: 90% examination 10% coursework Normal Pre-requisites: MATH0057 recommended Lecturer: Dr J Talbot Course Description and Objectives The course aims to introduce students to discrete mathematics, a fundamental part of mathematics with many applications in computer science and related areas. The course provides an introduction to graph theory and combinatorics, the two cornerstones of discrete mathematics. The course will be offered to third or fourth year students taking Mathematics degrees, and might also be suitable for students from other departments. There will be an emphasis on extremal results and a variety of methods. Recommended Texts B Bollobás, Modern Graph Theory (Springer); B Bollobás, Combinatorics (Cambridge Univer- sity Press). Detailed Syllabus − Binomial coefficients, convexity. Inequalities: Jensen's, AM-GM, Cauchy{Schwarz. Graphs, subgraphs, connectedness, Euler circuits, cycles, trees, bipartite graphs and other basic concepts. Vertex colourings. Graphs with large girth and large chromatic number. − Extremal graph theory: Dirac's theorem. Ore's theorem. Mantel's theorem. Turán's theorem (several proofs including probabilistic and analytic). Kövári–Sós{Turántheorem with applications to geometry. Erd}os{Stonetheorem. Stability. Andrásfai{Erd}os–Sós theorem. − Set Systems: Basic definitions; set systems and the discrete cube. Chains and antichains. Sperner's lemma. The LYM inequality. Intersecting families; the Erdos-Ko-Rado theorem (probabilistic and compression proofs). Isoperimetric problems: local LYM inequality, Kruskal{Katona theorem. The linear algebra method: Fisher's inequality, Ray- Chaudhuri{Wilson theorem. − Ramsey theory: Ramsey's theorem. Upper and lower bounds including probabilistic ideas. Schur's Theorem. Fermat's Last Theorem is false in Zp for any sufficiently large prime p. Van der Waerden's theorem on arithmetic progressions. May 2021 MATH0029.

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1. Directed Graphs Or Quivers What Is Category Theory? • Graph Theory on Steroids • Comic Book Mathematics • Abstract Nonsense • the Secret Dictionary
1. Directed graphs or quivers What is category theory? • Graph theory on steroids • Comic book mathematics • Abstract nonsense • The secret dictionary Sets and classes: For S = fX j X2 = Xg, have S 2 S , S2 = S Directed graph or quiver: C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) Class C0 of objects, vertices, points, . Class C1 of morphisms, (directed) edges, arrows, . For x; y 2 C0, write C(x; y) := ff 2 C1 j @0f = x; @1f = yg f 2C1 tail, domain / @0f / @1f o head, codomain op Opposite or dual graph of C = (C0;C1;@0;@1) is C = (C0;C1;@1;@0) Graph homomorphism F : D ! C has object part F0 : D0 ! C0 and morphism part F1 : D1 ! C1 with @i ◦ F1(f) = F0 ◦ @i(f) for i = 0; 1. Graph isomorphism has bijective object and morphism parts. Poset (X; ≤): set X with reﬂexive, antisymmetric, transitive order ≤ Hasse diagram of poset (X; ≤): x ! y if y covers x, i.e., x 6= y and [x; y] = fx; yg, so x ≤ z ≤ y ) z = x or z = y. Hasse diagram of (N; ≤) is 0 / 1 / 2 / 3 / ::: Hasse diagram of (f1; 2; 3; 6g; j ) is 3 / 6 O O 1 / 2 1 2 2. Categories Category: Quiver C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) with: • composition: 8 x; y; z 2 C0 ; C(x; y) × C(y; z) ! C(x; z); (f; g) 7! g ◦ f • satisfying associativity: 8 x; y; z; t 2 C0 ; 8 (f; g; h) 2 C(x; y) × C(y; z) × C(z; t) ; h ◦ (g ◦ f) = (h ◦ g) ◦ f y iS qq <SSSS g qq << SSS f qqq h◦g < SSSS qq << SSS qq g◦f < SSS xqq << SS z Vo VV < x VVVV << VVVV < VVVV << h VVVV < h◦(g◦f)=(h◦g)◦f VVVV < VVV+ t • identities: 8 x; y; z 2 C0 ; 9 1y 2 C(y; y) : 8 f 2 C(x; y) ; 1y ◦ f = f and 8 g 2 C(y; z) ; g ◦ 1y = g f y o x MM MM 1y g MM MMM f MMM M& zo g y Example: N0 = fxg ; N1 = N ; 1x = 0 ; 8 m; n 2 N ; n◦m = m+n ; | one object, lots of arrows [monoid of natural numbers under addition] 4 x / x Equation: 3 + 5 = 4 + 4 Commuting diagram: 3 4 x / x 5 ( 1 if m ≤ n; Example: N1 = N ; 8 m; n 2 N ; jN(m; n)j = 0 otherwise | lots of objects, lots of arrows [poset (N; ≤) as a category] These two examples are small categories: have a set of morphisms.
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