Questions: Consider the following graph: This graph... contains neither an Euler Circuit nor a Hamilton Cycle contains a Hamilton Cycle but not an Euler Circuit contains an Euler Circuit but not a Hamilton Cycle contains an Euler Circuit and a Hamilton Cycle

Consider the following graph:

This graph...
contains neither an Euler Circuit nor a Hamilton Cycle
contains a Hamilton Cycle but not an Euler Circuit
contains an Euler Circuit but not a Hamilton Cycle
contains an Euler Circuit and a Hamilton Cycle

Transcript text: Consider the following graph: This graph... contains neither an Euler Circuit nor a Hamilton Cycle contains a Hamilton Cycle but not an Euler Circuit contains an Euler Circuit but not a Hamilton Cycle contains an Euler Circuit and a Hamilton Cycle

Solution

Solution Steps

Step 1: Check for Euler Circuit

An Euler circuit requires that all vertices have an even degree (even number of edges connected to it). In the given graph, several vertices have an odd degree. For example, the top-left vertex has a degree of 3. Therefore, the graph does not contain an Euler Circuit.

Step 2: Check for Hamilton Cycle

A Hamilton cycle requires visiting each vertex exactly once and returning to the starting vertex. A possible Hamilton Cycle exists: Start at the bottom left vertex, move up, across the top, down on the right, then connect the central vertex and its three adjacent vertices not yet visited, finally returning to the starting vertex.

Final Answer

The graph contains a Hamilton Cycle but not an Euler Circuit.

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