Injective coloring of subclasses of chordal graphs

An injective k-coloring of a graph $G=(V,E)$ is a function $f:V\to \{1,2,\dots ,k\}$ such that for every pair of vertices u and v having a common neighbor, $f\left(u\right)\ne f\left(v\right)$. The injective chromatic number ${\chi }_i\left(G\right)$ of a graph G is the minimum k for which G admits an injective k-coloring. Given a graph G and a positive integer k, Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. Decide Injective Coloring Problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this result by proving that Decide Injective coloring Problem remains NP-complete for undirected path graphs, a proper subclass of chordal graphs. Moreover, we show that it is not possible to approximate the injective chromatic number of an undirected path graph within a factor of $n^{1/3-\epsilon }$ in polynomial time for every $\epsilon >0$ unless ZPP = NP. On the positive side, we prove that the injective chromatic number of an interval graph is either $\Delta \left(G\right)$ or $\Delta \left(G\right)+1$, where $\Delta \left(G\right)$ is the maximum degree of G. We also characterize the interval graphs having ${\chi }_i\left(G\right)=\Delta \left(G\right)$ and ${\chi }_i\left(G\right)=\Delta \left(G\right)+1$. As a consequence of this characterization, we obtain a linear time algorithm to find the injective chromatic number of an interval graph.