Two paper appear to imply collapse via coloring $P_5$-free graphs Found this from graphclasses.org.

Two papers give conflicting results for coloring $P_5$-free
graphs  which appear to imply $P=NP$.

From Polynomial-time algorithm for vertex k-colorability of P_5-free graphs


  Abstract. We give the first polynomial-time algorithm for coloring vertices of P5 -free graphs with k colors. This settles an open problem and generalizes several previously known results.


From Some new hereditary classes where graph coloring remains NP-hard, p 5


  ... Coloring is NP-hard in $2K_2$-free ... and $\{C_5,P_5\} \cup \ldots$-free


$2K_2-free \subset P_5-free$ and the other class contains $P_5$.

According to graphclasses, another reason for hardness is
clique cover on the complement (another paper), click +Details
for references.

Question:


  What is wrong with this seeming contradiction?
One of those papers is about k-colorability and the other of

those papers does not specify a constant number of colors.

(For a similar example, consider k-variable-SAT versus SAT.)