How to color the verteces of triangles in different colors?

Question

I have a directed graph $G=(V_n, E)$, where $V_n =\{1,2,..., n\}$ is the set of vertices and $E$ is the set of edges.

I have found the set of triangles which looks like:

  #   A    B    C
 1    7   21   24
 2    8   10   13
 3    2    8   13
 4   14   19   25
 5   18   20   24
 6   20   21   24
 7    7   20   21
 8    7   20   24
 9    5   20   24

Here $A$, $B$, $C$ are id's of the vertex in the $k$-th triangle.

I need to use a minimum number of colors to color the verticies of triangles in different colors.

My attempt is:

There are three situations for two triangles:

1) Triangles #1 (7,21,24) and #2 (8,10,13) don't have the common vertex and I can use two colors.

2) Triangles #5 (18,20,24) and #7 (7,20,21) have the common vertex 20 and I can use two colors.

3) Triangles #7 (7,20,21) and #8 (7,20,24) have the two common vertices 7, 20 and I can use two colors.

Question. Which is an algorithm one can use to solve this vertex coloring task?

Edit. The close task on the edge's set is here https://stackoverflow.com/questions/40432322/color-each-edge-in-triangle-igraph