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Undergraduate
By diego on June 8, 2012 | Updated Dec. 6, 2017

The cube of a connected graph is hamiltonian

Prove that the vertices of any connected graph \(G\) can be listed in a cyclic order so that the distance in \(G\) of every two consecutive vertices is at most \(3\). Moreover, show that this can be …
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Undergraduate
By Shiva Kintali on June 5, 2012 | Updated Dec. 6, 2017

Properties of Petersen Graph

Petersen Graph is the graph shown below : Prove the following properties of the Petersen Graph : It is not planar. It is strongly-regular. It has a Hamiltonian path but no Hamiltonian Cycle. It i…
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Graduate
By Shiva Kintali on June 12, 2012 | Updated Dec. 6, 2017

Hamiltonian cycles in odd graphs

Let \(G\) be a graph such that all vertices of \(G\) have odd degree. Let \(e\) be any edge of \(G\). Prove that there are an even number of Hamiltonian cycles using \(e\).
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Undergraduate
By Shiva Kintali on Sept. 28, 2013 | Updated Jan. 4, 2018

Hamiltonicity of Line Graphs

Let \(G=(V,E)\) be a simple undirected graph. The line graph \(L(G)\) of \(G\) is defined as follows: Each vertex in \(L(G)\) corresponds to an edge in \(G\), and two vertices are connected by an e…
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Undergraduate
By Shiva Kintali on May 19, 2013 | Updated Dec. 6, 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…
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