Other
Graph theory
- Course
- Institution
It helps you to understand briefly about the graph theory in discrete mathematics
[Show more]
Seller
Follow
gnanasekarg
Content preview
Graph TheoryBenny Sudakov
18 August 2016
,Acknowledgement
Much of the material in these notes is from the books Graph Theory by Reinhard Diestel and
Introduction to Graph Theory by Douglas West.
1
,Contents
1 Basic notions 4
1.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Graph isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The adjacency and incidence matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Special graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Walks, paths and cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.9 Graph operations and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Trees 10
2.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Equivalent definitions of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Cayley’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Connectivity 17
3.1 Vertex connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Edge connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 2-connected graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Menger’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Eulerian and Hamiltonian cycles 24
4.1 Eulerian trails and tours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Hamilton paths and cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Matchings 28
5.1 Real-world applications of matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Hall’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Matchings in general graphs: Tutte’s Theorem . . . . . . . . . . . . . . . . . . . . . . 31
6 Planar Graphs 34
6.1 Platonic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7 Graph colouring 38
7.1 Vertex colouring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.2 Some motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.3 Simple bounds on the chromatic number . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.4 Greedy colouring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.5 Colouring planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8 More colouring results 43
8.1 Large girth and large chromatic number . . . . . . . . . . . . . . . . . . . . . . . . . 44
8.1.1 Digression: the probabilistic method . . . . . . . . . . . . . . . . . . . . . . . 45
8.1.2 Proof of Theorem 8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.2 Chromatic number and clique minors . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.3 Edge-colourings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2
, 8.4 List colouring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
9 The Matrix Tree Theorem 52
9.1 Lattice paths and determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
10 More Theorems on Hamiltonicity 57
10.1 Pósa’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10.2 Tournaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
11 Kuratowski’s Theorem 61
11.1 Convex drawings of 3-connected graphs . . . . . . . . . . . . . . . . . . . . . . . . . 62
11.2 Reducing the general case to the 3-connected case . . . . . . . . . . . . . . . . . . . . 65
12 Ramsey Theory 68
12.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
12.2 Bounds on Ramsey numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
12.3 Ramsey theory for integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
12.4 Graph Ramsey numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
13 Extremal problems 73
13.1 Turán’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
13.2 Bipartite Turán Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller gnanasekarg. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $300.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
62890 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now