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Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Task assignment using Hall's Theorem
You are given a collection of tasks each of which must be assigned a time slot in the range {\(1,\dots,n\)}. Each task \(j\) has an associated interval \(I_j = [s_j,t_j]\) and the time slot assigned t…
Mathematics
Graph Theory
assignment
halls theorem
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Graduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Algebraic dual of graphs
Let \(G\) be a connected graph. An algebraic dual of \(G\) is a graph \(G'\) such that \(G\) and \(G'\) have the same set of edges, any cycle of \(G\) is a cut of \(G'\), and any cut of \(G\) is a cyc…
Mathematics
Graph Theory
matroids
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Undergraduate
By
Shiva Kintali
on Oct. 3, 2013 | Updated Jan. 4, 2018
Saturated vertices and Maximum matchings
Let \(S\) be a set of vertices saturated by a matching \(M\) in a graph \(G\). Prove that some maximum matching also saturates all of \(S\). Must the statement be true for every maximum matchin…
Mathematics
Graph Theory
matching
0
Undergraduate
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
Graph complement
Let \(G\) be an undirected graph without self-loops and multi-edges. The complement of graph \(G\) is a graph \(\overline{G}\) on the same vertices such that two vertices of \(\overline{G}\) are adjac…
Mathematics
Graph Theory
connectivity
graph complement
0
Undergraduate
By
Shiva Kintali
on May 31, 2013 | Updated Dec. 6, 2017
Characterizations of Eulerian graphs
It is well-known that a graph is Eulerian if and only if every vertex has even degree. Prove the following alternate characterization of Eulerian graphs Prove that if \(G\) is Eulerian and …
Mathematics
Graph Theory
eulerian graph
0
Undergraduate
By
Shiva Kintali
on June 30, 2012 | Updated Dec. 6, 2017
Coloring graphs with odd cycles
Prove that a graph with at most two odd cycles has chromatic number of at most 3. Let \(G\) be a graph where every two odd cycles have at least a vertex in common. We call such graphs nicely-odd grap…
Mathematics
Graph Theory
graph coloring
0
Undergraduate
By
Shiva Kintali
on Nov. 12, 2012 | Updated Dec. 6, 2017
Alice, Bob and children
Alice and Bob decide to have children until either they have their first girl or they have \(k \geq 1\) children. Assume that each child is a boy or girl independently with probability 1/2, and that …
Mathematics
Probability
expectation
0
Undergraduate
By
Shiva Kintali
on July 5, 2012 | Updated Dec. 6, 2017
Cycle in k-connected graphs
Prove that every \(k\)-connected graph (\(k > 1\)) on at least \(2k\) vertices has a cycle of length at least \(2k\).
Mathematics
Graph Theory
connectivity
0
Graduate
By
Shiva Kintali
on June 11, 2012 | Updated Dec. 6, 2017
Embedding complete bipartite graphs
Let \(S\) be an orientable surface of genus \(g \geq 0\). Prove that for every \(g \geq 0\) there exists an integer \(t\) such that \(K_{3,t}\) cannot be drawn on \(S\) without any crossings. What is…
Mathematics
Graph Theory
graph embedding
0
High School
By
Shiva Kintali
on June 1, 2013 | Updated Dec. 6, 2017
The Sixth Sense
In the following, you are allowed to put any mathematical symbols on the left-hand side of the \("="\) sign to make the left-hand side evaluate to \(6\). For example, \(2+2+2=6\). Do this for all the …
Puzzles
Puzzles
math puzzle
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