Subfield: Combinatorics


Question: There are $n \geqslant 3$ islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route.

After each year, the ferry company will close a ferry route between some two islands $X$ and $Y$. At the same time, in order to maintain its service, the company will open new routes according to the following rule: for any island which is connected by a ferry route to exactly one of $X$ and $Y$, a new route between this island and the other of $X$ and $Y$ is added.

Suppose at any moment, if we partition all islands into two nonempty groups in any way, then it is known that the ferry company will close a certain route connecting two islands from the two groups after some years. Prove that after some years there will be an island which is connected to all other islands by ferry routes.

Solution:
Initially, we pick any pair of islands $A$ and $B$ which are connected by a ferry route and put $A$ in set $\mathcal{A}$ and $B$ in set $\mathcal{B}$. From the condition, without loss of generality there must be another island which is connected to $A$. We put such an island $C$ in set $\mathcal{B}$. We say that two sets of islands form a network if each island in one set is connected to each island in the other set.

Next, we shall included all islands to $\mathcal{A} \cup \mathcal{B}$ one by one. Suppose we have two sets $\mathcal{A}$ and $\mathcal{B}$ which form a network where $3 \leqslant|\mathcal{A} \cup \mathcal{B}|<n$. This relation no longer holds only when a ferry route between islands $A \in \mathcal{A}$ and $B \in \mathcal{B}$ is closed. In that case, we define $\mathcal{A}^{\prime}=\{A, B\}$, and $\mathcal{B}^{\prime}=(\mathcal{A} \cup \mathcal{B})-\{A, B\}$. Note that $\mathcal{B}^{\prime}$ is nonempty. Consider any island $C \in \mathcal{A}-\{A\}$. From the relation of $\mathcal{A}$ and $\mathcal{B}$, we know that $C$ is connected to $B$. If $C$ was not connected to $A$ before the route between $A$ and $B$ closes, then there will be a route added between $C$ and $A$ afterwards. Hence, $C$ must now be connected to both $A$ and $B$. The same holds true for any island in $\mathcal{B}-\{B\}$. Therefore, $\mathcal{A}^{\prime}$ and $\mathcal{B}^{\prime}$ form a network, and $\mathcal{A}^{\prime} \cup \mathcal{B}^{\prime}=\mathcal{A} \cup \mathcal{B}$. Hence these islands can always be partitioned into sets $\mathcal{A}$ and $\mathcal{B}$ which form a network.

As $|\mathcal{A} \cup \mathcal{B}|<n$, there are some islands which are not included in $\mathcal{A} \cup \mathcal{B}$. From the condition, after some years there must be a ferry route between an island $A$ in $\mathcal{A} \cup \mathcal{B}$ and an island $D$ outside $\mathcal{A} \cup \mathcal{B}$ which closes. Without loss of generality assume $A \in \mathcal{A}$. Then each island in $\mathcal{B}$ must then be connected to $D$, no matter it was or not before. Hence, we can put $D$ in set $\mathcal{A}$ so that the new sets $\mathcal{A}$ and $\mathcal{B}$ still form a network and the size of $\mathcal{A} \cup \mathcal{B}$ is increased by 1 . The same process can be done to increase the size of $\mathcal{A} \cup \mathcal{B}$. Eventually, all islands are included in this way so we may now assume $|\mathcal{A} \cup \mathcal{B}|=n$.

Suppose a ferry route between $A \in \mathcal{A}$ and $B \in \mathcal{B}$ is closed after some years. We put $A$ and $B$ in set $\mathcal{A}^{\prime}$ and all remaining islands in set $\mathcal{B}^{\prime}$. Then $\mathcal{A}^{\prime}$ and $\mathcal{B}^{\prime}$ form a network. This relation no longer holds only when a route between $A$, without loss of generality, and $C \in \mathcal{B}^{\prime}$ is closed. Since this must eventually occur, at that time island $B$ will be connected to all other islands and the result follows.