Consider a semi-regular bipartite graph $C$ consisting of two parts $A$ (having each vertex of degree $\Delta$) and $B$ (having each vertex of degree $2$). Let its chromatic polynomial be $C(x)$. Now, consider connecting each of the vertices in the independent sets $A$ and $B$ such that the resulting subgraph $A'$ is $\Delta$ regular and the subgraph $B'$ is $\beta$ regular. Also consider that the maximum of chromatic numbers of $A'$ and $B'$ is $\chi$ and the chromatic polynomials of $A'$ and $B'$ are $A(x)$ and $B(x)$ respectively. Then, what is the chromatic polynomial of the new graph $A'\cup B'\cup C$ where $\cup$ denotes union of vertex and edge sets of the subgraphs.
I think it is of the form $r(x)A(x)B(x)$ where $r(x)$ has some relation with $C(x)$. Specifically, I think the chromatic number of the new graph would be at most $2$ more than $\chi$.
Note that the case when the bipartite graph is just a bridge (edge) reduces $r(x)$ to $\frac{x-1}{x}$ . This may be verified by the deletion-contraction principle. But, what if we have more than one connecting edge between $A$ and $B$. Any hints? Thanks beforehand.
