The satisfiability problem is NP-complete but there are subclasses where all the instances are satisfiable. For this, restrictions on the shape of the formula are made. Darman and D\"ocker show that the subclass MONOTONE $3$-SAT-($k$,1) with $k \geq 5$ proves to be NP-complete and pose the open question whether instances of MONOTONE $3$-SAT-(3,1) are satisfiable. This paper shows that all instances of MONOTONE $3$-SAT-(3,1) are satisfiable using the new concept of a color-structures.

翻译：可满足性问题是 NP 完全的，但存在某些子类，其中所有实例均是可满足的。为此，需对公式形式施加限制。Darman 与 Döcker 证明了子类 MONOTONE $3$-SAT-($k$,1)（其中 $k \geq 5$）是 NP 完全的，并提出了一个开放性问题：MONOTONE $3$-SAT-(3,1) 的实例是否可满足。本文利用全新的颜色结构概念，证明了所有 MONOTONE $3$-SAT-(3,1) 实例均为可满足的。