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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
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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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 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
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
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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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 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
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 Sept. 27, 2013 | Updated Jan. 4, 2018
Graph complement
Let \(G\) be an undirected graph without self-loops and multi-edges. The complement of graph \(G\) is a graph \(\overline{G}\) on the same vertices such that two vertices of \(\overline{G}\) are adjac…
Mathematics
Graph Theory
connectivity
graph complement
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