Chromatic polynomial and edge-chromatic number of certain graphs

Question

I have a few questions regarding the chromatic polynomial and edge-chromatic number of certain graphs.

1. For an empty graph, is the edge-chromatic number $0, 1$ or not well-defined? I've come up with reasons for each ($0$ since there aren't any edges to colour; $1$ because there's one way of colouring $0$ edges; not defined because there is no edge colouring of an empty graph) but I can't decide which argument is valid - is it more of a convention type thing?
2. There's a chromatic polynomial for a tree on $n$ vertices, so I was wondering if there was a chromatic polynomial for a bipartite graph on $n$ vertices (the chromatic number is $2$ since you can colour one part one colour, and the other part another, so I know $P_{G}(k)$ must have a factor of $(k-1)$)?

Any help would be much appreciated, thanks in advance!

Answer 1

1. I suppose the answer here is it depends. Let's view edge coloring a graph as vertex coloring its line graph (they are equivalent.) If $G$ is the empty graph, or graph with an empty edge set, then $L(G)$ is what some may call the null graph. The null graph is quite interesting in that it gives rise to puzzling questions such as yours, as well as paradoxical ones (is the null graph connected?) According to the linked Wikipedia page, the chromatic number of the null graph is $0$, and hence the chromatic index of the empty graph is $0$. I highly recommend you give $``$Is the null-graph a pointless concept?$"$ by Harary and Read a read if you can access it. It includes discussion on the chromatic polynomial of the null graph, which they determine is the constant function $1$, on the basis that $``$... no matter what the number $\lambda$ of colors, there is only one way to color the points of $K_0$ [the null graph], namely, do nothing!$"$ They also discuss the aforementioned connectedness paradox.

2. I don't quite understand the question here. Trees are bipartite, so are you asking if $k(k-1)^{n-1}$ is the chromatic polynomial of all bipartite graphs? Certainly not, and it may not be the case that there is a $``$standard form$"$ for chromatic polynomials of bipartite graphs. For example, even cycles have chromatic polynomial $P_{C_{2n}}(k) = (k-1)^{2n} + (k-1)$ and are chromatically unique. You may want to search for questions regarding the chromatic polynomial of the complete bipartite graph $K_{n,m}$, of which there are several.

To address your side note, any graph with an edge will have $k-1$ as a factor of its chromatic polynomial, as there is no way to properly color a graph with at least one edge with $1$ color.