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 ⟺ ∀A⊆X, |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 G−M (since it's (d−1)-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 (n−r) 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 A⊆X; 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=2n−1, 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: m≤3n−6 for n≥4. Proof: by Euler. An edge has at most 2 faces; face has at least 3 edges.
Bipartite planar graph: m≤2n−4. There are no odd cycles, so, any face has at least 4 edges.
2.3Platonic solids¶
Only 5. Proof: Euler's formula, using 2m=n⋅d (cus regular) and 2m=f⋅p (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(A∪B)=P(A)+P(B)−P(A∩B)
3.2Conditional¶
P(A|B)=P(A∩B)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!}