On-line graph coloring of $${\mathbb{P}_5}$$-free graphs

Kierstead et al. (SIAM J Discret Math 8:485–498, 1995) have shown 1 that the competitive function of on-line coloring for $${\mathbb{P}}_5$$ -free graphs (i.e., graphs without induced path on 5 vertices) is bounded from above by the exponential function $${\left( 4^{\chi (\mathbb{G})} - 1\right) / 3}$$ . No nontrivial lower bound was known. In this paper we show the quadratic lower bound $$\tiny{\left( {\begin{array}{*{20}c} {{\chi ({\mathbb{G}}) + 1}} \\ {2} \\ \end{array} } \right) }$$ . More precisely, we prove that $$\tiny{\left( {\begin{array}{*{20}c} {{\chi ({\mathbb{G}}) + 1}} \\ {2} \\ \end{array} } \right) }$$ is the exact competitive function for ($${\mathbb{C}}_4, {\mathbb{P}}_5$$)-free graphs. In this paper we also prove that 2$$\kappa({\mathbb{G})}$$ - 1 is the competitive function of the best clique covering on-line algorithm for ($${\mathbb{C}}_4, {\mathbb{P}}_5$$)-free graphs.