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Undergraduate
By
Shiva Kintali
on Jan. 21, 2013 | Updated Jan. 16, 2017
Characterizing outerplanar graphs
A graph is outerplanar if it is isomorphic to a plane graph such that every vertex is incident with the unbounded face. Prove that a graph is outerplanar if and only if it has no subgraph isomorph…
Mathematics
Graph Theory
planar graphs
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Undergraduate
By
TrueShelf Inc.
on Dec. 13, 2013 | Updated Jan. 16, 2017
Every edge in a triangle
For \(n \geq 3\), determine the minimum number of edges in a connected \(n\)-vertex graph in which every edge belongs to a triangle.
Mathematics
Graph Theory
triangles
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Undergraduate
By
Shiva Kintali
on Aug. 12, 2012 | Updated Jan. 16, 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
JeffE
on June 7, 2012 | Updated Jan. 16, 2017
Maintaining fitstrings
Every non-negative integer can be represented as the sum of distinct positive Fibonacci numbers. (As a warmup exercise, prove this claim!) In other words, instead of a string of bits, we can represe…
Computer Science
Algorithms
amortized analysis
0
Graduate
By
Shiva Kintali
on June 6, 2012 | Updated Jan. 16, 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
0
High School
By
Shiva Kintali
on July 18, 2012 | Updated Jan. 16, 2017
Red Blue Hats
There are 10 people and 10 hats. Each person is assigned a random hat, colored either RED or BLUE, but the number of each colored hat is not known to them. They will be lined up such that each person …
Puzzles
Puzzles
logic puzzle
0
Undergraduate
By
Shiva Kintali
on June 22, 2012 | Updated Jan. 16, 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
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
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
rajeshchitnis
on Aug. 2, 2012 | Updated Jan. 16, 2017
Numbers divisible by sum of their digits
Show that there is an infinite sequence of natural numbers \({p_1,p_2,\ldots}\) such that for every \(i\in \mathbb{N}\) the number \(p_i\) does not have 0 as any of its digits but is divisible by the …
Mathematics
Puzzles
Number Theory
Puzzles
divisibility
0
Undergraduate
By
aa1062
on July 22, 2012 | Updated Jan. 16, 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
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