A graph is said to be open-neighborhood conflict-free $k$-colorable if there exists an assignment of $k$ different colors to some of the vertices such that, for every vertex $v$, there is a color that is assigned to exactly one vertex among the neighbors of $v$. The open-neighborhood conflict-free chromatic number $\chi _O(G)$ is the smallest $k$ for which $G$ is open-neighborhood conflict-free $k$-colorable. Z. Abel et al., [ SIAM J. Discrete Math. 32 (2018), pp. 2675--2702] showed that $\chi _O(G)\leq 8$ for every planar graph $G$ and posed two open problems to ask whether $\chi _O(G)\leq 4$ for every planar graph $G$ and whether $\chi _O(G)\leq 3$ for every outerplanar graph $G$. We present in this paper positive answers for the above two open problems by establishing a stronger result which states that, for every integer $k\ge 2$, every minor-$k$-colorable graph is open-neighborhood conflict-free $k$-colorable, where a graph $G$ is said to be minor-$k$-colorable if every minor of $G$ is $k$-colorable.

In this paper, we consider the two-stage scheduling problem in which n jobs are first processed on m identical machines at a manufacturing facility and then delivered to their customers by one vehicle which can deliver one job at each shipment. In the problem, a set of n delivery times is given in advance, and in a schedule, the n delivery times should be assigned to the n jobs, respectively. The objective is to minimize the maximum delivery completion time, i.e., the time when all jobs are delivered to their respective customers and the vehicle returns to the facility. For this problem, we present a 32-approximation algorithm and a polynomial-time approximation scheme.