A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in the literature. We prove that a (dart, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or is the join or co-join of two smaller graphs. Using this structure result, we design a polynomial-time algorithm for finding an optimal colouring of (dart, odd hole)-free graphs. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $\omega^2$ where $\omega$ denotes the number of vertices in a largest clique of $G$. We prove that (dart, odd hole)-free graphs are perfectly divisible.

翻译：洞是指顶点数至少为四的无弦圈。若一个洞的顶点数为奇数，则称其为奇洞。dart 是一个包含顶点 $a, b, c, d, e$ 及边 $ab, bc, bd, be, cd, de$ 的图。dart-自由图在文献中已被广泛研究。我们证明：一个(dart, odd hole)-自由图或是完美图，或是不包含一个由三个顶点构成的稳定集，或是两个更小图的联图或共联图。利用这一结构结果，我们设计了一个多项式时间算法，用于求解(dart, odd hole)-自由图的最优着色问题。若图 $G$ 的每个导出子图 $H$ 都包含一个顶点集 $X$，使得 $X$ 与 $H$ 的所有最大团相交，且 $X$ 导出一个完美图，则称图 $G$ 是完全可分的。完全可分图 $G$ 的色数以 $\omega^2$ 为上界，其中 $\omega$ 表示 $G$ 中最大团的顶点数。我们证明了(dart, odd hole)-自由图是完全可分的。