Complexity of the conditional colorability of graphs

For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different colors. The smallest integer $k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called the $r$th order conditional chromatic number, denoted by $\chi_r(G)$. It is easy to see that the conditional coloring is a generalization of the traditional vertex coloring for which $r=1$. In this paper, we consider the complexity of the conditional colorings of graphs. The main result is that the conditional $(3,2)$-colorability is $NP$-complete for triangle-free graphs with maximum degree at most 3, which is different from the old result that the traditional 3-colorability is polynomial solvable for graphs with maximum degree at most 3. This also implies that it is $NP$-complete to determine if a graph of maximum degree 3 is $(3,2)$- or $(4,2)$-colorable. Also we have proved that some old complexity results for traditional colorings still hold for the conditional colorings.