# coloring the vertices of the following graph with 3 colors

find the number of way that we can coloring the vertices of the following graph with 3 colors : red,white and green such that no two adjacent vertices are of the same color; this is called a vertex coloring.

i think we have $$3!$$ ways for right above triangle and $$2 \times 2$$ ways for left above triangle so answer is $$3! \times 2 \times 2 =24$$ .is this true ?

• As you said and as commented by Morgan, among the $2 \times 2$ ways of coloring the left triangles, in one case you can choose the color of the middle vertex, therefore, the number of possible colorings is $30$. Commented Jan 4, 2023 at 8:01
• @VezenBU: That looks like an answer to me, not a comment. Commented Jan 4, 2023 at 10:23
• Can the graph rotate in 3D or 2D ,because we may need to apply Burnsides' lemma in that situations Commented Jan 4, 2023 at 16:13

As you said and as commented by Morgan, among the $$2 \times 2$$ ways of coloring the left triangles, in one case you can choose the color of the middle vertex, therefore, the number of possible colorings is $$30$$ instead of $$24$$.