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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 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
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
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
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
0
Undergraduate
By
Shiva Kintali
on June 5, 2012 | Updated Dec. 6, 2017
Properties of Petersen Graph
Petersen Graph is the graph shown below : Prove the following properties of the Petersen Graph : It is not planar. It is strongly-regular. It has a Hamiltonian path but no Hamiltonian Cycle. It i…
Mathematics
Graph Theory
connectivity
hamiltonian cycle
matching
petersen graph
0
Graduate
By
Shiva Kintali
on July 28, 2012 | Updated Dec. 6, 2017
Petersen’s theorem
Prove that every 2-edge-connected 3-regular graph has a perfect matching
Mathematics
Graph Theory
connectivity
matching
0
Graduate
By
Shiva Kintali
on July 2, 2012 | Updated Dec. 6, 2017
Recognizing bipartite k-extendable graphs
A graph \(G\) is \(k\)-extendable, where \(k \geq 1\) is an integer, if every matching of size at most \(k\) is a subset of a perfect matching of \(G\). Given a bipartite graph \(G\) and an integer …
Computer Science
Mathematics
Algorithms
Graph Theory
matching
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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