Genus of a simple graph

Question

Let $G$ is a finite simple undirected graph. Suppose there exist subgraph $G_1,G_2,\dots,G_n$ of $G$, such that $G_i \cong K_5$ or $K_{3,3}$, $E(G_i)\cap E(G_j) = \emptyset$ and $|V(G_i)\cap V(G_j)| \leq 3$, for $i\neq j$. Then, is it true that genus of $G$ is greater than or equal to $n$?

Thanks in advance.

Answer 1

Second try. I believe $K_{6,3}$ is a counterexample.

It is genus $1$ and contains $2$ edge disjoint $K_{3,3}$s sharing only $3$ vertices.

Explicitly:

K_{6,3}=[(0, 6), (0, 7), (0, 8), (1, 6), (1, 7), (1, 8), (2, 6), (2, 7), (2, 8), (3, 6), (3, 7), (3, 8), (4, 6), (4, 7), (4, 8), (5, 6), (5, 7), (5, 8)] first K_{3,3}=[(3, 6), (3, 7), (3, 8), (4, 6), (4, 7), (4, 8), (5, 6), (5, 7), (5, 8)] second K_{3,3}=[(0, 6), (0, 7), (0, 8), (1, 6), (1, 7), (1, 8), (2, 6), (2, 7), (2, 8)] shared vertices 8,6,7 

Added $K_{3,3n}$ are counterexamples too.

$g(K_{3,3n})=\lceil (3n-2)/4 \rceil < (3n-2) / 4 + 1$

$K_{3,3n}$ has $n$ disjoint $K_{3,3}$ sharing only $3$ vertices: connect the $3$ partition to $3$ distinct vertices from the $3n$ partition.