  A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the
vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get
precisely two colors. We study graphs $G$ which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a $K_3$-WORM coloring with two
colors. In fact for every integer $k\ge 3$ there exists a $K_3$-WORM colorable
graph in which the minimum number of colors is exactly $k$. There also exist
$K_3$-WORM colorable graphs which have a $K_3$-WORM coloring with two colors
and also with $k$ colors but no coloring with any of $3,\dots,k-1$ colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide $k$-colorability for every $k \ge 2$ (and remains
intractable even for graphs of maximum degree 9 if $k=3$). On the other hand,
we prove positive results for $d$-degenerate graphs with small $d$, also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.
