Basic Graph Theory Problem Set II for RMO and INMO

  1. An Eulerian trail in a digraph is a trail containing all the edges. An Eulerian circuit is a closed trail containing all the edges. Show that a digraph X contains an Eulerian circuit if and only if d^{+}(v)=d^{-}(v) for every vertex v and the underlying graph has at most one component.
  2. Determine for what values of m \ge 1 and n \ge 1 is K_{m,n} Eulerian.
  3. What is the maximum number of edges in a connected, bipartite graph of order n?
  4. How many 4-cycles are in K_{m,n}?
  5. Let Q_{n} be the n-dimensional cube graph. Its vertices are all the n-tuples of 0 and 1 with two vertices being adjacent if they differ in precisely one position. Show that Q_{n} is connected and bipartite.

More later,

Nalin Pithwa

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