Chromatic index of a complete graph Looking to show that $\forall n \in \mathbb{N}$
$$\chi^{'}(K_{2n+2})=\chi^{'}(K_{2n+1})=2n+1$$
I'm trying to construct a colouring of the edges of $K_{2n+1}$ that leaves colour $i$ missing at vertex $i$ so I can move to $K_{2n+2}$ without increasing the index (for my induction) but the details of that initial colouring are hard to work out.
 A: I agree with Leen that induction might not be the way to go.  But still, most proofs on the colorability of $K_n$ do not construct an explicit coloring.  I think it's worth it to construct one at least once, so here goes :)
Say your vertex set is $V = \{0, 1, \ldots, n - 1\}$ with $n$ odd.  
Color the edge $\{i, j\}$  with color $i + j$ (mod $n$).  You should be able to show that no vertex has two incident edges of the same color.
Now, one needs to show that each vertex in this coloring misses a distinct color.  This'll show that $K_{n + 1}$ can be colored with $n$ colors, $n + 1$ being even.
Actually, vertex $i$ misses color $2i$ (mod $n$).  For otherwise, there are distinct $i, j$ such that $i + j \equiv 2i$ (mod $n$).  This implies that either $j = i$ or $j = n + i$, the two of which are not possible.  All that remains for you to show is that for distinct $i, j$, $2i \not \equiv 2j$ (mod $n$), and hence that all vertices have a distinct color they're missing.
A: You probably should not use induction.
Vizing tells you that $\chi'(K_k)$ is either $k-1$ or $k$. If $\chi(K_k)=k-1$ every color appears at every vertex, so the edges of one color form a perfect matching, which implies that $k$ must be even.
You said you could handle the other part? (The other part is proved if you know, or can prove, that for odd $k$ $K_k$ has a decomposition in Hamiltonian cycles. Let me know if you don't know how to do this. It is not hard.)
A: Just another solution: Label the vertices $1,2,\ldots, 2n+1$ and for each $i$, colour edge $(i+k,i-k)$ using colour $i$ for every $k=1,2,\ldots,n$ (operations modulo $2n+1$). This is a proper edge colouring using $2n+1$ colours with the property that no edge coloured $i$ is incident to vertex $i$. 
Example for $2n+1=7$:

