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Things to know for the final CC-BY-NC

Maintainer: admin

1Graph matchings

1.1Stable marriage

Gale-Shapley: male choice method, male-optimal, female-pessimal (lemma: no woman leaves/rejects a valid partner)

1.2Matching cardinality

Match M is maximal no augmenting path for M. Proof: () just switch M and not M in the path. () Symmetric diff with M and M (a theoretical maximal matching); all vertices have deg 2, so there are only disjoint paths and cycles. Specifically, we have even-length paths, even-length cycles, and odd-length paths ending in M - but the latter is augmenting wrt M.

1.3Bipartite graphs

No odd cycles. Can be separated into 2 groups, each edge spans both.

1.3.1Hall's theorem

Bipartite graph has a PM AX, |Γ(A)||A|. Proof: sort of by induction on X, it's weird

1.3.2Regular graphs

d-regular: all vertices have degd. Any d-reg bipartite graph can be decomposed into d edge-disjoint PMs. Proof by induction: if it has a PM M, can apply IH to GM (since it's (d1)-regular) and then throw M back in there. To prove it has a PM, show that Hall's holds.

1.3.3Latin rectangles

Extending to a square: X = columns, Y = numbers. Edge between column and number if the number is not in the column. Results in a (nr) regular graph, decomposes into PMs, make those into new rows.

1.3.4Edge colouring

Bipartite graph: χE(G)=Δ(G) (max deg); proof by adding dummy edges until it's d-regular and then decomposing

1.3.5Systems of distinct representatives

SDR exists union of any sets size of any one set. Proof by Hall's.

1.3.6The house-selling problem

Bipartite graph. Find a perfect matching. If you can't, find an unsuppliable set AX; increase house prices in Γ(A) and decrease house prices of undemanded houses; continue until PM is found.

1.3.7Telecommunication networks

Crossbar: N inputs/outputs; N2 connections, O(N4) crossovers.

Clos: r n×m, then m r×r, then r m×n. Fewer connections, fewer crossovers. If m=2n1, all packets can be routed online. For offline routing, create a bipartite graph, edge between (i,πi), use Hall's.

1.3.8König-Egerváry

Largest matching = smallest vertex cover

2Planar graphs

2.1Euler's formula

m+2=n+f. Proof by induction: base case = tree; otherwise, cycle - remove an edge and apply IH.

2.2Other formulas

Any planar graph: m3n6 for n4. Proof: by Euler. An edge has at most 2 faces; face has at least 3 edges.

Bipartite planar graph: m2n4. There are no odd cycles, so, any face has at least 4 edges.

2.3Platonic solids

Only 5. Proof: Euler's formula, using 2m=nd (cus regular) and 2m=fp (cus an edge has 2 faces).

2.4Vertex-colouring

Six-colourability proof: using Euler's, min degree <6; remove, apply IH, add back.

2.5Art gallery theorem

n walls n/3 guards. Proof: split up floor into triangle, prove 3-colourability by induction.

2.6Planar duals

Swap faces and vertices.

2.7Graph minors

Contracting edges is basically all you need. Equivalent to contracting trees.

2.8Kuratowski's theorem

Planar no K3 or K5,5 minor. Proof by minimum counterexample.

2.9Fáry's theorem

Any planar graph has a straight line drawing. Proof by induction?

2.10Menger's theorem

Max flow min cut basically

3Probability

3.1Union and intersection

P(AB)=P(A)+P(B)P(AB)

3.2Conditional

P(A|B)=P(AB)P(B)

3.3Bayes' theorem

P(A|B)=P(B|A)P(A)P(B)

3.4Discrete random variables

Linearity of expectation: E(aX+bY)=aE(X)+bE(Y)

For independent events: E(XY)=E(X)E(Y)

3.5Binomial distribution

Bernouilli trial: success with prob p.

\binom{n}{k} = \frac{n!}{k!(n-k)!}

3.6Random walks

Markov chain model. Just matrix multiplication. Stationary distribution with d_v/2m for any v.

3.7Markov's inequality

P(Z > c \cdot E(Z)) \leq \frac{1}{c}

3.8Chernoff bounds

Ind. Bernouilli trials X = X_1 + X_2 + \ldots + X_j, \mu = E(X). Then:

P(X > (1+\delta)\mu) \leq \left ( \frac{e^\delta}{(1+\delta)^{(1+\delta)}} \right )^\mu

Proof: let f(x) = e^{tx} for some t, apply to both sides of inequality of LHS. Use Markov's, simply with p. Use the fact that 1+x \leq e^x (or 1-x \leq e^{-x} for the < inequality). Take the deriv of the exponent, set to 0 to find t.

Corollary: 1 > \delta > 0, P(X > (1+\delta)\mu) \leq e^{-1/3\mu\delta^2}. Proof by Taylor series.

Another: b \geq 6 \mu, P(X > b) \leq 2^{-b}. Proof since 2e < 6.

4Combinatorics

4.1Calatan numbers

+-+-+ etc with each partial sum (from the left) being positive.

C(n) = \frac{1}{n+1} \binom{2n}{n}

Proof: Using D(n), the reflection thing, etc.

4.2Generating functions

4.2.1Ordinary

For counting labelled objects

F(x) = \sum_{n \geq 0} f(x) x^n

4.3Exponential

For counting unlabelled objects

F(x) = \sum_{n \geq 0} f(x) \frac{x^n}{x!}

4.4Binomial theorem

(1+x)^n = \sum_{k=0}^n \binom{n}{k} x^k

Generalised: let \displaystyle \binom{n}{k} = \frac{n \cdot (n-1)\cdots (n-k+1)}{k!}