Exact Complexity of Exact-Four-Colorability

Let $M_k \seq \nats$ be a given set that consists of $k$ noncontiguous integers. Define $\exactcolor{M_k}$ to be the problem of determining whether $\chi(G)$, the chromatic number of a given graph $G$, equals one of the $k$ elements of the set $M_k$ exactly. In 1987, Wagner \cite{wag:j:min-max} proved that $\exactcolor{M_k}$ is $\bhlevel{2k}$-complete, where $M_k = \{6k+1, 6k+3, >..., 8k-1 \}$ and $\bhlevel{2k}$ is the $2k$th level of the boolean hierarchy over $\np$. In particular, for $k = 1$, it is DP-complete to determine whether $\chi(G) = 7$, where $\DP = \bhlevel{2}$. Wagner raised the question of how small the numbers in a $k$-element set $M_k$ can be chosen such that $\exactcolor{M_k}$ still is $\bhlevel{2k}$-complete. In particular, for $k = 1$, he asked if it is DP-complete to determine whether $\chi(G) = 4$. In this note, we solve this question of Wagner and determine the precise threshold $t \in \{4, 5, 6, 7\}$ for which the problem $\exactcolor{\{t\}}$ jumps from NP to DP-completeness: It is DP-complete to determine whether $\chi(G) = 4$, yet $\exactcolor{\{3\}}$ is in $\np$. More generally, for each $k \geq 1$, we show that $\exactcolor{M_k}$ is $\bhlevel{2k}$-complete for $M_k = \{3k+1, 3k+3,..., 5k-1\}$.