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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
0
Undergraduate
By
Shiva Kintali
on May 22, 2013 | Updated Jan. 16, 2017
Basics of bipartite graphs
Prove the following properties of bipartite graphs : Prove that a graph \(G\) is bipartite if and only if every subgraph \(H\) of \(G\) has an independent set consisting of at least half of \(V(H)\)…
Mathematics
Graph Theory
basics
bipartite graph
0
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 May 19, 2013 | Updated Jan. 16, 2017
Large min-degree implies perfect matching
Let \(G\) be a bipartite graph with partitions \(X\) and \(Y\) such that \(|X|=|Y|=n\). The degree of each vertex in \(G\) is at least \(n/2\). Prove that \(G\) has a perfect matching.
Mathematics
Graph Theory
matching
0
Undergraduate
By
Shiva Kintali
on May 9, 2012 | Updated Jan. 16, 2017
$K_4$ subdivision and 3-colorability
Prove that every graph with no subgraph isomorphic to a subdivision of \(K_4\) is 3-colorable.
Mathematics
Graph Theory
graph coloring
0
Undergraduate
By
rizwanhudda
on July 1, 2012 | Updated Jan. 16, 2017
Top K elements in a read only array
Given a read only array of \(n\) integers, find the \(k\) largest numbers in the array in \(O(n)\) time possibly using \(O(k)\) extra space. EDIT: Note \(k\) is not a constant, so when I say \(O(n)\)…
Computer Science
Algorithms
linear time algorithms
0
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 May 19, 2013 | Updated Jan. 16, 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
Undergraduate
By
Shiva Kintali
on May 31, 2013 | Updated Jan. 16, 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 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
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