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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
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
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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
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Graduate
By
Shiva Kintali
on June 11, 2012 | Updated Jan. 16, 2017
Embedding complete bipartite graphs
Let \(S\) be an orientable surface of genus \(g \geq 0\). Prove that for every \(g \geq 0\) there exists an integer \(t\) such that \(K_{3,t}\) cannot be drawn on \(S\) without any crossings. What is…
Mathematics
Graph Theory
graph embedding
0
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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Graduate
By
Shiva Kintali
on June 7, 2012 | Updated Jan. 16, 2017
Planar Graphs are Pfaffians
Let \(G(V,E)\) be an undirected graph. An orientation of \(G\) is Pfaffian if every even cycle \(C\) such that \(G \setminus V(C)\) has a perfect matching, has an odd number of edges directed in eithe…
Mathematics
Graph Theory
pfaffian
planar graphs
0
Graduate
By
Shiva Kintali
on Feb. 20, 2013 | Updated Jan. 16, 2017
Treewidth and Cliques
A tree decomposition of a graph \(G(V, E)\) is a pair \(\mathcal{D} = ({X_i\ |\ i \in I}, T(I, F))\) where \({X_i\ |\ i \in I}\) is a collection of subsets of \(V\) (called bags) and \(T(I, F)\) is a …
Mathematics
Graph Theory
treewidth
0
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
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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
0
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 …
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