This might come across as a slightly petty question. Apologies for this, I am only asking as I have an exam on Graph Theory soon and want to make sure I do things correctly.
The definition of a Hamilton path is a path which includes every vertex of the graph exactly once.
The definition of a circuit is a path whose first and last vertices are the same.
A Hamilton circuit is a Hamilton path which is a circuit.
Question - isn't this a contradiction (it is supposed to include every vertex exactly once but then include the initial vertex twice)? What exception on these rules is brought in so it isn't contradictory?
Furthermore, if I wanted to quote a Hamilton circuit in the undirected and unweighted cyclic graph G, with vertices a,b,c where edges are { (a,b) , (b,c) , (c,a) } - would I quote the Hamilton circuit as abca or abc ?
Thanks for any help on this.
It seems like your problem is less with Hamiltonian circuits and more with circuits in general; you are right that that is a contradiction. The problem, however, is your definition of circuit. What you have is a closed walk. My definition of a circuit is a closed walk that repeats no edges. Unfortunately, this is a problem with graph theory in that the terminology is not codified, though I don't know of anyone calling a closed walk a circuit. This is something you'll need to check for yourself based on what your instructor teaches, either in your text, or notes.
I can give you the definitions with which I'm familiar, and hope they mesh with the ones you know. How I define a Hamiltonian circuit (though I prefer cycle), and incidentally, how Wikipedia does too, is a closed walk where each vertex occurs once and only once, except for the first and last, which are the same. The idea is that a Hamiltonian circuit is a Hamiltonian path up to the second to last vertex, and then it returns to the start. You are right, however, that that would not be a path.
I'm not sure what you mean by quoting a cycle, but if you just want to present one, the least confusing way of doing so would be as you just did: $\{(a,b), (b,c), (c,a)\}$. Again, you should check with your instructor in case you're expected to follow some sort of format, but that is a way of presenting one. I would, however, suggest you give it as a sequence rather than a set, e.g. $(a,b), (b,c), (c,a)$. Even though if a set of edges give a Hamiltonian cycle, there is only one way of interpreting it, it just feels more intuitive to me to present the edges in the order in which they occur.