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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
Graduate
By
Shiva Kintali
on Sept. 11, 2012 | Updated Dec. 6, 2017
Vertex cover in bipartite graphs
The Vertex Cover of a graph \(G(V,E)\) is a set of vertices \(S \subseteq V\) such that each edge of the graph is incident to at least one vertex of the set \(S\). Minimum cost vertex cover : Given a…
Mathematics
Optimization
Graph Theory
Linear Programming
bipartite graph
vertex cover
0
Undergraduate
By
Shiva Kintali
on July 30, 2012 | Updated Dec. 6, 2017
k-regular bipartite graphs are 2-connected
Prove that every connected \(k\)-regular bipartite graph on at least three vertices is 2-connected.
Mathematics
Graph Theory
bipartite graph
connectivity
0
Undergraduate
By
Shiva Kintali
on Sept. 29, 2013 | Updated Jan. 4, 2018
Fibonacci numbers and primes
The Fibonacci numbers are defined by \(F_1 = F_2 = 1\) and \(F_n = F_{n−1} + F_{n−2}\) for \(n \geq 3\). If \(p\) is a prime number, prove that at least one of the first \(p + 1\) Fibonacci numbers mu…
Mathematics
Combinatorics
fibonacci
pigeonhole principle
primes
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
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Regular bipartite super-graph
Let \(G\) be a simple bipartite graph where each side of the bi-partition has size \(n\). The maximum degree of \(G\) is \(\Delta \le n/10\). Show that there exists a \(2 \Delta\)-regular simple bip…
Computer Science
Mathematics
Algorithms
Graph Theory
bipartite graph
graph algorithms
0
Undergraduate
By
Shiva Kintali
on May 21, 2013 | Updated Dec. 6, 2017
Covering complete graph with bipartite graphs
The union of graphs \(G_1, \dots, G_k\), written \(G_1 \cup \dots \cup G_k\), is the graph with vertex set \(V(G_1) \cup \dots \cup V(G_k)\) and edge set \(E(G_1) \cup \dots \cup E(G_k)\). Prove tha…
Mathematics
Graph Theory
bipartite graph
graph union
0
Graduate
By
Shiva Kintali
on Nov. 15, 2012 | Updated Dec. 6, 2017
Basics of Perfect Graphs
A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Prove that bipartite graphs are perfect. Prove that the lin…
Mathematics
Graph Theory
basics
bipartite graph
chordal graph
graph coloring
interval graph
perfect graphs
0
Undergraduate
By
Shiva Kintali
on June 17, 2012 | Updated Dec. 6, 2017
Matching saturating high degree vertices
Let \(G\) be a bipartite multigraph and let \(\Delta\) be its maximum degree. Prove that \(G\) has a matching saturating every vertex of degree \(\Delta\).
Mathematics
Graph Theory
bipartite graph
matching
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