The Wikipedia article on Equitable coloring states that
A polynomial time algorithm is known for equitable coloring of split graphs.
The referred paper also seems to achive the proposed polynomial time, as far as my understanding goes.
One the other hand some other published paper states the following:
EQUITABLE COLORING of the disjoint union of split graphs parameterized by the number of colors is $W$-hard.
Where the disjoint union is defined as follows: The disjoint union of two graphs $G\cup H$ is a graph such that $V (G\cup H) = V (G)\cup V (H)$and $E(G \cup H) = E(G) \cup E(H)$.
One the one hand a poly-time algorithm seems like a great result for split graphs, on the other hand we have a $W$-hard problem for the disjoint union of such graphs. This seems odd to me as the disjoint union is only a couple of split graphs combined to one bigger graph (in which itself each split graph isnt even connected).
As far as my understanding goes we could just color each individual split-graph of the union with the poly-time algorithm and we would still achive some optimal result in polynomial time.
What am I missing here? What is the exact difference that leads to this "jump" in complexity? And most importantly what is wrong in my "attempt" to get a poly-time algorithm for equitable coloring "disjoint union of split graphs"?