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Undergraduate
By
Shiva Kintali
on June 29, 2012 | Updated Jan. 16, 2017
Coloring Planar Graphs
Prove that every planar graph has at least one vertex of degree at most five. Conclude that planar graphs are six-colorable. Prove that every triangle-free planar graph has at least one vertex of deg…
Mathematics
Graph Theory
eulers formula
planar graphs
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Undergraduate
By
JeffE
on June 6, 2012 | Updated Jan. 16, 2017
Longest increasing digital subsequence
Let \(S[1..n]\) be a sequence of integers between \(0\) and \(9\). A digital subsequence of \(S\) is a sequence \(D[1..k]\) of integers such that each integer \(D[i]\) is the numerical value of a sub…
Computer Science
Algorithms
dynamic programming
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Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Jan. 16, 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
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 …
Puzzles
Puzzles
strategy
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Undergraduate
By
Shiva Kintali
on July 5, 2012 | Updated Jan. 16, 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 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
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
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
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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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
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