Confusion colouring path graphs I'm a bit confused about how to count the number of colourings on a path graph. In my textbook, the following theorem is given:
In general, if $G$ is a path on $n$ vertices, then $P(G,\lambda)=\lambda(\lambda - 1)^{n-1}$
Where $\lambda$ is the number of colours available. Obviously, this is the number of proper colourings of $P_n$ (the path graph on $n$ vertices), where no two adjacent vertices have the same colour.
Am I correct that this also assumes that the graph is labelled? If the graph weren't labelled then the colouring $RGB$ would be the same as the colouring $BGR$ and shouldn't be counted twice.
This brings me to my main question. I'm being asked to count the number of colourings of $P_n$ with the colours red, green, and blue, such that no two adjacent vertices are blue. I'm confused as to whether I should be assuming that $P_n$ is labelled or not? I know the theorem from the text isn't exactly relevant to my question but it assumes that $P_n$ is labelled, but my understanding of paths untill now had been that they're to be taken as unlabelled.
I'm looking for advice on how to interpret and begin to approach this problem, and hopefully clear up some of my confusion. Thanks
 A: You should always assume that a graph is labeled. An "unlabeled graph" isn't really a graph; it's an equivalence class of graphs with respect to isomorphism. (Or some other definition that cashes out to the same thing.)
If this clashes with statements you've encountered like "A clique in a graph is a $K_n$ subgraph for some $n$" - well, the precise way to say that is "A clique in a graph is a subgraph which is a copy of $K_n$" - in other words, "a subgraph which is isomorphic to $K_n$". I'm not entirely sure what you mean when you say "my understanding of paths until now had been that they're to be taken as unlabelled" but it's probably something along those lines.
In particular, you should assume that a graph is labeled when counting colorings. A coloring of a graph $G$ is a function $V(G) \to C$, where $C$ is some set of colors. A proper coloring is a coloring in which no two adjacent vertices are colored (that is, mapped to) the same color.
If $G$ is a path on $n$ vertices and $|C|=\lambda$, then there are $\lambda^n$ possibly-improper colorings, because that's how many functions there are from an $n$-element set to a $\lambda$-element set. Of those colorings, $\lambda (\lambda-1)^{n-1}$ are proper.
