I have a graph made from two subgraphs: complete bi-parted $K_{4,4}$ and two triangles graph 
. These two subgraphs are connected by two edges. My task is to prove the number of proper 3-colorings for this graph. I can prove 3-colorings for $K_{4,4}$ and for two triangles, jet I cannot prove once they are connected together. Is there a way to make a "formal" proof for something like this?
Edit:
I was trying to prove it by multiplying the number of proper colorings and then dividing it be the number of incorrect combinations. There are 36 combinations for connection, but 18 are incorrect, so I divided the result of multiplication it by 2. The number is correct, jet it's not a correct proof.
Let the node in the "northeast corner" of $K_{4,4}$ be $k_1$ and the node in the "southeast corner" be $k_2.$ Let the node in the "northwest corner" of the two-triangle graph be $t_1$ and the "southwest corner" be $t_2.$ In any proper coloring of $K_{4,4},$ $k_1$ and $k_2$ get different colors. Let us say $k_1$ is colored red and $k_2$ is colored blue. In any coloring of he two-triangle graph, $t_1$ and $t_2$ must get different colors, say $X$ and $Y$ respectively. The two colorings are compatible if and only if $X$ is not red and $Y$ is not blue.
Note that for any proper coloring $C$ of the two-triangle graph, there is a proper coloring $C'$ obtained by exchanging the colors of $t_1$ and $t_2.$ Also, $C\neq C'$ since $X\neq Y.$ Since only three colors are available one of $X,Y$ must be red or blue, so not both of $C,C'$ are compatible with a given coloring of $K_{4,4}.$ But one of them must be, for the colors of $k_i$ and $t_i$ are the same in $C$ if and only if they are different in $C'$ for $i1,2.$ Thus the compatible colorings of the two-triangle graph are in one-to-one correspondence with the incompatible colorings, so half the colorings are compatible with a given coloring of $K_{4,4}.$
This argument seems to be what you are looking for, though I would actually argue somewhat differently. I would say that if $k_1, k_2$ are colored are colored red and blue respectively, then the colors assigned to $t_1, t_2$ can only be (blue, red), (blue, green), or (green, red). Then the color of the central node in the two-triangle graph is determined, and there are two ways to assign colors to the rightmost two nodes, giving $3\cdot2=6$ compatible colorings.