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Undergraduate
By
diego
on June 27, 2012 | Updated Dec. 6, 2017
Unmatchable edges of bipartite graphs
Prove that the following algorithm finds the unmatchable edges of a bipartite graph \(G\) (edges that aren't in any perfect matching): find a perfect matching in \(G\), orient the unmatched edges from…
Mathematics
Graph Theory
bipartite graph
matching
0
Undergraduate
By
123forman
on May 8, 2014 | Updated Dec. 6, 2017
Democratic distribution of wealth
A crew of 100 pirates have captured 100 laptops. The pirates are ranked 1 through 100, with pirate 100 being the pirate king. Captured booty is distributed as follows. The pirate king proposes a dis…
Mathematics
Discrete Mathematics
Game Theory
induction
0
Graduate
By
Shiva Kintali
on Aug. 8, 2012 | Updated Dec. 6, 2017
Gallai Identities
Consider the following parameters of an undirected graph \(G\) on \(n\) vertices. \(\nu(G)\) is the size of a maximum matching of \(G\). \(\tau(G)\) is the size of a minimum vertex cover of \(G\). …
Mathematics
Graph Theory
edge cover
independent set
matching
vertex cover
0
Undergraduate
By
Shiva Kintali
on June 8, 2012 | Updated Dec. 6, 2017
Linear time algorithms on trees
Let \(T(V,E)\) be a tree. Design linear time (i.e., \(O(|V|)\) time) algorithms for the following problems : Find an optimal vertex cover in \(T\). Find a maximum matching in \(T\). Find a maximum i…
Computer Science
Mathematics
Algorithms
Graph Theory
independent set
linear time algorithms
matching
trees
vertex cover
0
High School
By
Shiva Kintali
on June 27, 2013 | Updated Jan. 4, 2018
Basics of Induction
Prove the following using induction: \(\sum_{i=1}^{n}{i} = \frac{n(n+1)}{2}\). \(\sum_{i=1}^{n}{i}^2 = \frac{n(n+1)(2n+1}{6}\). \(\sum_{i=1}^{n}{i}^3 = {(\frac{n(n+1)}{2})}^2\). …
Mathematics
Discrete Mathematics
induction
summation
0
Graduate
By
Shiva Kintali
on June 9, 2012 | Updated Dec. 6, 2017
Minimum edge cover vs Maximum matching
An edge cover of a graph \(G(V,E)\) is a subset \(F \subseteq E\) of edges such that every node is incident to at least one edge in \(F\). Show that a minimum cardinality edge cover can be determine…
Mathematics
Optimization
Graph Theory
Linear Programming
edge cover
matching
reduction
1
Undergraduate
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
Fibonacci numbers and Induction
The Fibonacci numbers, \(F_0, F_1, F_2, \dots\) , are defined recursively by the equations \(F_0 = 0\), \(F_1 = 1\), and \(F_n = F_{n-1} + F_{n-2},\) for \(n > 1\). Prove that …
Mathematics
Discrete Mathematics
fibonacci
induction
tiling
0
Graduate
By
Shiva Kintali
on Sept. 23, 2012 | Updated Dec. 6, 2017
Characterizing factor-critical graphs
A graph \(G\) is said to be factor-critical if \(G-v\) has perfect matching for every \(v \in V(G)\). Prove that no bipartite is factor-critical. Show that \(G\) is factor-critical if and only if …
Mathematics
Graph Theory
bipartite graph
matching
0
High School
By
Shiva Kintali
on Sept. 30, 2013 | Updated Jan. 4, 2018
Using Binomial Theorem
Prove the following using binomial theorem and/or mathematical induction Let \(a\), \(b\) and \(n\) be natural numbers, prove that \(\frac{(a+\sqrt{b})^n + (a-\sqrt{b})^n}{2}\) is also a natural …
Mathematics
Discrete Mathematics
binomial theorem
induction
0
Graduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Perfect Matchings in Cubic Graphs
Let \(G\) be a cubic graph with no cut-edge. Let \(PM(G)\) be the convex hull of the perfect matchings of \(G\). Prove that the vector \((1/3, 1/3, \dots, 1/3) \in PM(G)\). Prove that every two-conn…
Mathematics
Optimization
Combinatorial Optimization
Graph Theory
matching
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