Originally, without a direct plan or pattern, I began testing each graph, seeing if each of them were traversable or not and recorded the results of each graph on a piece of paper like this: 1) Traversable, 2) etc. After this, I started examining the graphs that were not traversable in order to see a pattern as to why they were untraversable. For example, Graph 4 was untraversable and it had 5 nodes and 8 edges. Instantly, I came up with the hypothesis that 'if there is an odd number of nodes, the graph is not traversable' however, upon further inspection, I found that some of the graphs that were traversable also had and odd number of nodes.
After my first virtual defeat, I moved on to the next untraversable graph which turned out to be Graph 5. In this graph, there were eight nodes (further excluding my previous theory) and 12 edges. As one would expect, I found that both of these graphs had an equal number of sides. Continuing with this hypothesis, I examined the other traversable graphs to see if any of them had an even number of sides as well which would automatically cancel my theory. Almost immediately, in Graph 1, I noticed that it had also an even number of sides. I was back to square one once again.
After a few thought-provoking minutes and a couple of false theories, it finally struck me! Since the beginning, I had only considered the number of nodes and edges. But what if I combined these variables? What if I counted the number of edges connecting to each node? Once more, one-by-one, I examined each Graph, calculating the number of edges connecting to each node. Finally I concluded with this:
Graph 1: 2 nodes with an odd number of edges connected to it (traversable)
Graph 2: 0 nodes with an odd number of edges connected to it (traversable)
Graph 3: 0 nodes with an odd number of edges connected to it (traversable)
Graph 4: 4 nodes with an odd number of edges connected to it (untraversable)
Graph 5: 8 nodes with an odd number of edges connected to it (untraversable)
Graph 6: 4 nodes with an odd number of edges connected to it (untraversable)
Graph 7: 2 nodes with an odd number of edges connected to it (traversable)
Graph 8: 0 nodes with an odd number of edges connected to it (traversable)
Graph 9: 4 nodes with an odd number of edges connected to it (untraversable)
Graph 10: 2 nodes with an odd number of edges connected to it (traversable)
Finally, after all that work, I realised that the graphs that were traversable had a number of 0 or 2 odd nodes. Therefore, this meant that the untraversable graphs had a number other than 0 or 2 of odd nodes.
:)