By Alspach B., Xu M.Y.

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C) For which values of n, is the n-cube Eulerian? 28. A representation of Q4 13) Let n ≥ 1. 50) is one. Compare this graph with the n-cube introduced above. 40 Advanced Graph Theory and Combinatorics 14) For the n-cube, determine its edge-connectivity λ(Qn ). 15) Prove that a (multi)graph G is bipartite if and only if every circuit in G has an even length. 16) Find all values a, b, c with 1 ≤ a ≤ b ≤ c such that the complete tripartite graph Ka,b,c has a Eulerian trail but no Eulerian circuit. 17) Give examples of simple graphs such that λ(G) = i for i = 1, 2, 3, 4.

The conclusion follows from lines 10–12 of the algorithm. 6. g. Google’s PageRank, graphs associated with social networks like Facebook or Twitter, collaboration graphs and computing shortest path for a GPS device. g. electrical distribution systems, water running in a series of pipes of different diameters with various capacities and computer networks and distributed resources. We can also think about quivers17 occurring in the study of friezes in algebraic combinatorics [BER 16, Chapter 10]. Let us present six more examples.

Second, given a Hamiltonian graph G, if we also provide a candidate for a Hamiltonian circuit, it is easy to check that the graph is indeed Hamiltonian. The size of the certiﬁcate is less than the size of the graph: a circuit is a subgraph of G. We can easily produce a veriﬁcation algorithm to check Hamiltonicity of the given circuit: is the provided list of edges a circuit? Is every vertex of the graph visited exactly once? These questions have to be answered positively. In view of this example, if a problem belongs to NP, it seems maybe challenging – or at least computationally challenging – to decide whether an instance is positive.