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Undergraduate
By
Shiva Kintali
on June 26, 2012 | Updated Jan. 16, 2017
Complete balanced binary trees are graceful
Label the vertices of a tree (with \(n\) vertices) with integers from \(1\) to \(n\). Now label each edge with the absolute difference of the labels of its incident vertices. The labeling is said to b…
Mathematics
Graph Theory
graph labeling
0
Undergraduate
By
domotorp
on June 18, 2012 | Updated Jan. 16, 2017
Volley planning
Suppose there are n soldiers in a row who want to fire at the same time. Each of them has a constant memory and can only communicate with its neighbors, i.e. send message to left or right one and rece…
Puzzles
Puzzles
communication complexity
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Undergraduate
By
Shiva Kintali
on June 25, 2012 | Updated Jan. 16, 2017
Separator number of a tree
Let \(T(V,E)\) be a tree with \(|V| = n\) vertices. Let \(W\) be a subset of \(V\). Prove that there is a vertex \(v \in V\) such that every component of \(T - v\) contains at most …
Computer Science
Mathematics
Algorithms
Graph Theory
trees
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Undergraduate
By
aa1062
on July 22, 2012 | Updated Jan. 16, 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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Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Jan. 16, 2017
Basics of Hamiltonicity
Let \(G(V,E)\) be a simple graph with \(n\) vertices and minimum degree \(\delta\). Also, for every two vertices \(x, y \in V\), \(|N(x) \cup N(y)| + \delta \ge n + 10\). Prove that \(G\) has a Hami…
Mathematics
Graph Theory
basics
hamiltonian cycle
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Graduate
By
Shiva Kintali
on June 8, 2012 | Updated Jan. 16, 2017
Simplicial vertices in Chordal graphs
Let \(G(V,E)\) be a simple undirected graph. Let \(v \in V\) and let \(N(v)\) denote the neighbors of \(v\) in \(G\). The vertex \(v\) is said to be simplicial if \(N(v)\) induces a clique in \(G\). …
Mathematics
Graph Theory
chordal graph
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
0
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
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
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