On color-critical ($P_{5},\overline{P}_5$)-free graphs
  A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical $P_5$-free graphs for each $k \geq 5$. We show that the number of $k$-critical $(P_5, \overline{P}_5)$-free graphs is finite for every fixed $k$. Our result implies the existence of a certifying algorithm for $k$-coloring $(P_5, \overline{P}_5)$-free graphs. 
Harjinder S. Dhaliwal, Ang\`ele M. Hamel, Ch\'inh T. Ho\`ang,   Fr\'ed\'eric Maffray, Tyler J. D. McConnell, Stefan A. Panait
math.CO