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Graduate
By
Shiva Kintali
on June 4, 2013 | Updated Dec. 6, 2017
Treewidth and feedback vertex set
A feedback vertex set is a set of vertices whose removal results in an acyclic graph. Prove that if a graph has a feedback vertex set of size \(k\), then it has treewidth at most \(k+1\).
Mathematics
Graph Theory
treewidth
0
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 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
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
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
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
0
Undergraduate
By
Shiva Kintali
on May 9, 2012 | Updated Dec. 6, 2017
$K_4$ subdivision and 3-colorability
Prove that every graph with no subgraph isomorphic to a subdivision of \(K_4\) is 3-colorable.
Mathematics
Graph Theory
graph coloring
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
True Putnam
on Aug. 28, 2013 | Updated Jan. 4, 2018
Putnam 2005 A6
Let \(n\) be given, \(n \geq 4\), and suppose that \(P_1,P_2, \dots,P_n\) are \(n\) randomly, independently and uniformly, chosen points on a circle. Consider the convex \(n\)-gon whose vertices are \…
Mathematics
Probability
uniform distribution
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Connectivity of dual graph
Prove that if \(G\) is a simple 3-connected planar graph with at least 4 vertices, then the dual of \(G\) is also a simple 3-connected planar graph.
Mathematics
Graph Theory
planar graphs
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