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Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Minimum of Random Subsets
Let \(S = \){\(1,2,\dots,n\)}. Let \(A,B\) be two random subsets of \(S\). Let \(\min(A)\) denote the minimum number in the set \(A\). What is the probability that \(\min(A)= \min(B)\) ? Evalua…
Mathematics
Probability
probability
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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
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Undergraduate
By
Shiva Kintali
on Aug. 22, 2013 | Updated Jan. 4, 2018
Graphs and Fermat's Little Theorem
Given a prime number \(n\), let \(\mathbb{Z}_n\), denote the set of congruence classes of integers modulo \(n\). Let \(a\) be a natural number having no common prime factors with \(n\); multiplication…
Mathematics
Graph Theory
Number Theory
digraphs
fermats little theorem
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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 May 19, 2013 | Updated Dec. 6, 2017
Self-complementary graphs
Let \(G\) be a self-complementary graph (i.e., \(G\) is isomorphic to its complement) on \(n\) vertices. Prove that \(n \equiv 0\ (mod\ 4)\) or \(n \equiv -1\ (mod\ 4)\). Prove that \(G\) has a cut…
Mathematics
Graph Theory
graph complement
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Undergraduate
By
aa1062
on July 22, 2012 | Updated Dec. 6, 2017
Prisoners finding numbers
A prison contains \(n\) prisoners, labeled \(1, 2, 3, \dots, n\). One day the warden announces that he is going to set up a room with \(n\) drawers in it, labeled \(1, 2, 3, \dots, n\). He will then …
Puzzles
Puzzles
strategy
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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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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
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
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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