Consider a complete multipartite graph $G$ with $k$ $(k\ge2)$ parts with equal number of vertices $n$, where $n$ is even. If $k=2$, it is a well known result that the Alon-Tarsi number (ATN)(The maximum exponent of a variable in a monomial having least degree among all monomials of the graph polynomial) of $G$ equals $\frac{n}{2}+1$. Actually, this formula is derived from the formula for average degree $\frac{E(G)}{V(G)}$, which is a lower bound on $ATN$ of any graph. For complete bipartite graphs, this bound is always achieved.
My question is, whether the same holds for complete multipartite graphs. Like, my logic is, as any two partite sets of the $k$ parts form a complete bipartite graph, the subgraph polynomial of these complete bipartite graphs will have a nonzero monomial with exponent $\frac{n}{2}$. So, in the full graph polynomial, by multiplying all the non-zero monomials having exponent $\frac{n}{2}$ of the corresponding complete bipartite graphs formed by any two parts would give us a monomial having exponent $(k-1)\frac{n}{2}$, thus making $ATN(G)$ to achieve the lower bound. What could possibly prevent this to happen? Any hints?
