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Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a partition of the polygon into $K$ polygonal (but not necessarily convex) parts whose interiors are pairwise disjoint, $$ \mathcal{Z}_K(P)=\{p_1,\ldots,p_K\}\qquad\text{such that}\qquad \operatorname{int}(p_i) \cap \operatorname{int}(p_j)=\emptyset\;\forall i\neq j\quad\wedge\quad \bigcup_{i=1}^K p_i=P. $$

Question. Is it true for every $K$, that there is such a partition $\mathcal{Z}_K(P)$ that allows all parts $p_i$ to share at least one edge or vertex with every other part $p_{j\neq i}$?

I tried for $n=4$ and $K=2,3,4$ and succeeded (trivial), but I got stuck at $K=5$. I tried to search for this specific partitioning but I couldn't find anything relevant.

I only found out that if we let $k_i$ be the centroid of the i-th part $p_i$, and connect such centroid to all other centroids $k_{j\neq i}$ of polygonal parts that share at least one edge or vertex with every other part, the resulting graph will have an all-ones adjacency matrix.

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a partition of the polygon into $K$ polygonal (but not necessarily convex) parts whose interiors are pairwise disjoint, $$ \mathcal{Z}_K(P)=\{p_1,\ldots,p_K\}\quad\text{s.t.}\quad \operatorname{int}(p_i) \cap \operatorname{int}(p_j)=\emptyset\;\forall i\neq j\quad\wedge\quad \bigcup_{i=1}^K p_i=P. $$

Question. Is it true for every $K$, that there is such a partition $\mathcal{Z}_K(P)$ that allows all parts $p_i$ to share at least one edge or vertex with every other part $p_{j\neq i}$?

I tried for $n=4$ and $K=2,3,4$ and succeeded (trivial), but I got stuck at $K=5$. I tried to search for this specific partitioning but I couldn't find anything relevant.

I only found out that if we let $k_i$ be the centroid of the i-th part $p_i$, and connect such centroid to all other centroids $k_{j\neq i}$ of polygonal parts that share at least one edge or vertex with every other part, the resulting graph will have an all-ones adjacency matrix.

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a set of the $K$ polygon vers (vers are dual to covers) of $P$, namely $$ \mathcal{Z}_K(P)=\{p_1,\ldots,p_K\}\qquad\text{such that}\qquad p_i \cap p_j=\emptyset\;\forall i\neq j\quad\wedge\quad \bigcup_{i=1}^K p_i=P $$

where $p_i$ are polygonal (not necessarily convex) parts of $P$. In addition, also the interiors of $p_i$ are pairwise disjoint and $\mathcal{Z}_K(P)$ is not the set of all possible $K$ partitions of $P$, but a single partition into $K$ parts.

Question. Is there a partition $\mathcal{Z}_K(P)$ that allows all parts $p_i$ to share at least one edge or vertex with every other part $p_{j\neq i}$, for all $K$? If not, are there any $K$ for such partition is impossible?

I tried for $n=4$ and $K=2,3,4$ and succeeded (trivial), but I got stuck at $K=5$. I tried to search for this specific partitioning but I couldn't find anything relevant.

I only found out that if we let $k_i$ be the centroid of the i-th part $p_i$, and connect such centroid to all other centroids $k_{j\neq i}$ of polygonal parts that share at least one edge or vertex with every other part, the resulting graph will have an all-ones adjacency matrix.

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a partition of the polygon into $K$ polygonal (but not necessarily convex) parts whose interiors are pairwise disjoint, $$ \mathcal{Z}_K(P)=\{p_1,\ldots,p_K\}\qquad\text{such that}\qquad \operatorname{int}(p_i) \cap \operatorname{int}(p_j)=\emptyset\;\forall i\neq j\quad\wedge\quad \bigcup_{i=1}^K p_i=P. $$

Question. Is it true for every $K$, that there is such a partition $\mathcal{Z}_K(P)$ that allows all parts $p_i$ to share at least one edge or vertex with every other part $p_{j\neq i}$?

I tried for $n=4$ and $K=2,3,4$ and succeeded (trivial), but I got stuck at $K=5$. I tried to search for this specific partitioning but I couldn't find anything relevant.

I only found out that if we let $k_i$ be the centroid of the i-th part $p_i$, and connect such centroid to all other centroids $k_{j\neq i}$ of polygonal parts that share at least one edge or vertex with every other part, the resulting graph will have an all-ones adjacency matrix.

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a set of the $K$ polygon vers (vers are dual to covers) of $P$, namely $$ \mathcal{Z}_K(P)=\{p_1,\ldots,p_K\}\qquad\text{such that}\qquad p_i \cap p_j=\emptyset\;\forall i\neq j\quad\wedge\quad \bigcup_{i=1}^K p_i=P $$

where $p_i$ are polygonal (not necessarily convex) parts of $P$. In addition, $\mathcal{Z}_K(P)$ is not the set of all possible $K$ partitions of $P$, but a single partition into $K$ parts.

Question. Is there a partition $\mathcal{Z}_K(P)$ that allows all parts $p_i$ to share at least one edge or vertex with every other part $p_{j\neq i}$, for all $K$? If not, are there any $K$ for such partition is impossible?

I tried for $n=4$ and $K=2,3,4$ and succeeded (trivial), but I got stuck at $K=5$. I tried to search for this specific partitioning but I couldn't find anything relevant.

I only found out that if we let $k_i$ be the centroid of the i-th part $p_i$, and connect such centroid to all other centroids $k_{j\neq i}$ of polygonal parts that share at least one edge or vertex with every other part, the resulting graph will have an all-ones adjacency matrix.

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a set of the $K$ polygon vers (vers are dual to covers) of $P$, namely $$ \mathcal{Z}_K(P)=\{p_1,\ldots,p_K\}\qquad\text{such that}\qquad p_i \cap p_j=\emptyset\;\forall i\neq j\quad\wedge\quad \bigcup_{i=1}^K p_i=P $$

where $p_i$ are polygonal (not necessarily convex) parts of $P$. In addition, also the interiors of $p_i$ are pairwise disjoint and $\mathcal{Z}_K(P)$ is not the set of all possible $K$ partitions of $P$, but a single partition into $K$ parts.

Question. Is there a partition $\mathcal{Z}_K(P)$ that allows all parts $p_i$ to share at least one edge or vertex with every other part $p_{j\neq i}$, for all $K$? If not, are there any $K$ for such partition is impossible?

I tried for $n=4$ and $K=2,3,4$ and succeeded (trivial), but I got stuck at $K=5$. I tried to search for this specific partitioning but I couldn't find anything relevant.

I only found out that if we let $k_i$ be the centroid of the i-th part $p_i$, and connect such centroid to all other centroids $k_{j\neq i}$ of polygonal parts that share at least one edge or vertex with every other part, the resulting graph will have an all-ones adjacency matrix.