Graph Theory

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?

There exist graphs for which no matter the nodes the edge is attached to, yes For a certain pair of graphs connected by an edge attached to certain nodes, yes Regardless of the graphs and the nodes to which the edge is added, yes No, it is impossible

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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Graph Theory - Eulerian Paths

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