Find a decomposition of K2n into two subgraphs G1 and G2 such that χ(G1) + χ(G2) = n+2.

Question

I've found a decomposition for $K_4$ into 2 $P_2$'s and a $C_4$ and several decompositions of $K_6$ and above that fit this property where $\chi(G)$ is the chromatic number of the given graph. How would this generalize to $K_{2n}$ for $n \geq 2, n \in \mathbb{N}$ ?

Answer 1

If the subgraphs don't need to be connected you can decompose $K_{2n}$ into $2K_n$ and $K_{n,n}$ with chromatic numbers $n$ and $2$ respectively. If they do need to be connected, remove one edge from the second subgraph and add it to the first.

Answer 2

Label the vertices $u_1,v_1,\ldots,u_n,v_n$. Let one subgraph be the cycle $u_1v_1\cdots u_nv_nu_1$ with chromatic number $2$. The other subgraph $G_2$ consists of all other edges.

Now $G_2$ contains all edges $u_ju_k$ and hence has chromatic number at least $n$. Also there is an $n$-colouring where $u_j$ and $v_j$ have colour $j$ (because $u_j$ and $v_j$ are not adjacent in $G_2$). So $G_2$ has chromatic number exactly $n$.