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Undergraduate
By
aa1062
on July 22, 2012 | Updated Dec. 6, 2017
Pecking order
A researcher is studying the social dynamics of chicken coops. In each coop, for each pair of chickens \(A\) and \(B\), there is a pecking relationship: either \(A\) pecks \(B\) or \(B\) pecks \(A\) (…
Mathematics
Graph Theory
tournament
0
Undergraduate
By
Shiva Kintali
on July 17, 2012 | Updated Dec. 6, 2017
Number of triangles in a graph
Prove that a simple graph with \(n\) vertices and \(m\) edges has at least \(\frac{m}{3n}(4m − n^2)\) triangles.
Mathematics
Graph Theory
counting
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
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Edge disjoint perfect matchings
Let \(G\) be a \(d\)-regular bipartite graph with \(n\) vertices in each partition. Prove that \(G\) can be decomposed into \(d\) edge disjoint perfect matchings.
Mathematics
Graph Theory
matching
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 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
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 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 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
0
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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