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Discrete Mathematics

# Graph Theory - Eulerian Paths

What must be true of a path that is an Eulerian path?

True or false, if a graph has an Eulerian path then it has an Eulerian circuit.

True or false, if $$G$$ is a connected graph with at least two nodes, then an Eulerian path in $$G$$ must visit every node.

Two graphs $$G$$ and $$G'$$ each have at least one Eulerian circuit. Let $$G''$$ be a graph derived by combining both $$G$$ and $$G'$$ with a single edge between some node in $$G$$ to some node in $$G'$$. Does $$G''$$ have an Eulerian circuit?

Suppose a connected graph has 15 nodes. Given that is has an Eulerian circuit, what is the minimum number of distinct Eulerian circuits which it must have?

NOTE: A circuit uses an ordered list of nodes, so a circuit with nodes 1-2-3 is considered distinct from a circuit with nodes 2-3-1.

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