Course Content
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FLOWS
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Maximal and Minimal and Perfect Matchings
o Not important but stated in class anyway:
 (Compare 0s and 1s and say if you can get to 0s or 1s by changing them)
 ( 0,1,1,0,1,1) (0,1,0,0,1,0) -> (1,0,1,1,1,1)
Alternating and Augmenting Paths
o No matching edges in the end points, allows for you to increase the size
Hall’s Theorem
o When bipartite have perfect matchings
A matching is maximum <=> no augmenting paths (Bipartite case)
Tutte’s theorem
o General graphs
o If set of verts, then if you pull out, the size of the odd must be …..
o had assignment question on this
Networks (directed graphs with capacity function and source and sink)
Flows -> Functions f:E -> [0, infinity) satisfying conservation of flow and feasibility.
Maximum Flows always exist.
Augmenting paths
o
A flow is maximum <=> no augmenting paths to t
 <=> st separator of capcity val(f)
Max Flow min cut theorem
o Val of a max flow = cap of min separator
Flow integrality theorem
Mengers Theorem
PLANAR GRAPHS
 Euler’s formula : |V|-|E|+|F| = 2
 Bounds on number of edges of planar graphs
 |V| ≤ 3|E| - 6
 |V| ≤ 2|E| - 4 (for planar)
o Proved both from Euler and double counting and hand shake lemma
 Double counting, since each edge touches two faces, and you will count them
twice.
 Jordan Curve Theorem
 Kuratowski Theorem
o
G planar <=> no K5 or K3,3 subdivision
COLOURING
 Vertex X(G), Edge (X’(G) List, Xl(G)
o w(G) ≤ X(G) ≤ Δ(G) + 1
 Brookes Theorem (for connected G)
o X(G) ≤ Δ(G) unless G is a complete graph
o Δ(G) = 2 G is an odd cycle
 Greedy Coloring
o X(G) ≤ col(G) + 1
o Color one of least degree.
o X(G)≥ |V| / α(G)
 Since coloring is like a partitioning into independent sets.
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o
 Edge Coloring
o X’(G) ≥ Δ(G)
o X’(G) ≤ 2 Δ(G) – 1
 Since coloring in order, and take an edge ??
 Vizing Theorem
o X’(G) ≤ Δ(G) + 1
 Didn’t prove this
 Koning’s Theorem
o X’(G) = Δ(G) if G is bipartite
 Proved by successively removing matchings, and then for general, consider
embedding in another graph who’s ??? Increases..
 List Coloring
o Xl(G) ≤ Δ(G) + 1
o Xl(G) ≤col (G) + 1
o Example of bipartite graphs with large list chromatic #
 Colouring Planar Graphs
o Planar graphs always have a vertex of degree ≤ 5.
 So col(G) ≤ 5
 So X(G) ≤ 6
o There exists planar graph with X(G) ≥ 4
 Theorem (Thomasesen) If G is planar then Xl(G)≤5
o Complex induction. If you have embedding of G, and have property that all lists of
vertices of length at least 5 all vertices have length 3, then some vertices list have length
at least 1 so you can properly color from that list.
l
 X (G) = smallest k st wherever each vertex v is assigned a list Lv of colours |Lv|≥k there is a
proper colouring of G from the lists.
Trees
 Charactersized trees
o Unique path between any two vertices.
o Connected but removing any vertex disconnects
o No cycles but adding an edge creates a cycle
o Connected an |V| = |E|+1
o Has no cycles , |V| = |E|+1
 End vertex lemma
o A tree has at least two leaves if it has at least 2 verts
 Leaf Removal lemma
o Connected graph with some leaf, then graph is a tree only if removing a leaf gives a tree.
 Spanning Trees
o Algorithm K
 Adds edges one by one unless doing so creates a cycle.
 Input is a list of edges, ins ome order, and output is a spanning tree of G.
 Outputs the “first” spanning tree with respect to the input order.
o Kuruskal Algorithm
 G=(V,E) : w:E -> R
 Order the edges in non decreasing order of weight
 Run Algorithm K on edges in this order
Probability
 Proofs by counting
o Showed that in 4 shuffles you can’t get all permutations
o (if compliment isn’t everything, then it cant be nothing etc)
o That is the whole idea of proof by counting, and same as probability
 Axiom
o P(A U B ) ≤ P(A)+P(B) (union bound)
 Independence
o E={1 is a fixed point of a random permutation}
o F={---------}
o E and F not indep.
 Random variables
o Indicator functions
 Random set S, N = |
 N=sum(I[v in S])
 Linearity of expectation
o E[X1 + X2…. +Xk] = E[X} + …………. + Ek
 Markoves Inequality P(X≥t)≤EX/t
 Applications
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o Large bipartite subgraphs
o Turan’s theorem. Used in compliment form.
Ramsey Theorem
o Ramsety # r(k,l)
o r(k)= r(k,k)
o Upper bound r(k,l)≤(k+l-2 chose k-1) r(k,k_<= 4k-1)
 Proved by induction
o Lower bound
 r(k,k) >= 2k/2
o Note:
 r(k,l) >= r(k-1, l)
 r(k,l) >= r(k,l-1)
 if k>l then
 r(k,l) >= r(k-1, l) ….. >= r(l,l)
 if k<l then
 r(k,l) >= r(k,l-1) >= …. R(k,k)
 we get that
 r(k,l) >= min (r(k,k) , r (l,l)
 r( k,l) >= min(k,l) min (k,l)/2
Best thing to do is to understand solutions to assignments
Especially those without long proofs