Let $N$ be a positive integer,$G$ be a simple graph and $H_1,H_2,\ldots,H_k$ be a family of subgraphs of $G$ which satisfy:
- every $H_i$ is a $N$-order complete graph;
- the union of $H_i$ is $G$;
- the maximal clique size of $G$ is $N$;
- $H_i$ is not equal to $H_j$ and the intersection of $H_i$ and $H_j$ is not empty for any different $i$ and $j$;
- the intersection of $H_i$ is empty.
I want to ask what is the minimum possible value of the order of G.
(Here for any two simple graph $G_1$=($V_1$,$E_1$) and $G_2$=($V_2$,$E_2$),define the union of $G_1$ and $G_2$ be (the union of $V_1$ and $V_2$ , the union of $E_1$ and $E_2$), the intersection of $G_1$ and $G_2$ be (the intersection of $V_1$ and $V_2$ , the intersection of $E_1$ and $E_2$).)