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Undergraduate
By
Shiva Kintali
on June 22, 2012 | Updated Dec. 6, 2017
100 perfect logicians
100 perfect logicians are told to sit in a circle in a room. Before they enter the room, they are told at least one person has a blue forehead. When you determine your forehead is blue, you need to le…
Puzzles
Puzzles
logic puzzle
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Undergraduate
By
Shiva Kintali
on Aug. 12, 2012 | Updated Dec. 6, 2017
Finding perfect matching in bipartite graphs
Let \(G = (A,B)\) be a bipartite graph. Hall's theorem implies that \(G\) has a perfect matching if and only if \(|A| = |B|\) and for each \(X \subseteq A\), \(|X| \leq |\Gamma(X)|\), where …
Computer Science
Algorithms
matching
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Undergraduate
By
Shiva Kintali
on May 22, 2013 | Updated Dec. 6, 2017
Balls and bin game
Consider the following balls-and-bin game. We start with one black ball and one white ball in a bin. We repeatedly do the following : choose one ball from the bin uniformly at random, and then put the…
Mathematics
Probability
sampling
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Undergraduate
By
Shiva Kintali
on June 14, 2012 | Updated Dec. 6, 2017
Connectivity of cubic graphs
Prove that if \(G\) is 3-regular graph, then its vertex-connectivity equals its edge-connectivity.
Mathematics
Graph Theory
connectivity
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Partitioning planar graphs
Let \(G=(V,E)\) simple planar graph. Prove that \(V\) can be partitioned into three disjoint sets \(V = V_1 \cup V_2 \cup V_3\), such that the induced subgraphs on \(V_1\), \(V_2\) and \(V_3\) are acy…
Mathematics
Graph Theory
acyclic
graph partition
planar graphs
0
Undergraduate
By
Shiva Kintali
on June 10, 2012 | Updated Dec. 6, 2017
Expanding expressions
You are given an algebraic expression \(E\) having variables, addition, multiplication and parenthesis. Your goal is to repeatedly expand \(E\) using the distributive law if possible. Prove that th…
Puzzles
Puzzles
convergence
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Undergraduate
By
Shiva Kintali
on Oct. 3, 2013 | Updated Jan. 4, 2018
Eulerian facts
State whether each of the following statements are TRUE or FALSE. Your answers should be accompanied by a proof. Every Eulerian bipartite graph has an even number of edges. Every Eulerian simpl…
Mathematics
Graph Theory
eulerian graph
true or false
0
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
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 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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