All instances of MONOTONE 3-SAT-(3,1) are satisfiable

Abstract

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.