Explain and answer this question---4. The six offices of the city of Salavaara are to be connected to each other by a communication network which utilizes modern picotechnology. Each of the offices is to be connected to all the other ones by direct cable connections. Three operators compete to build the connections, and there is a separate competition for every connection. When the network is finished one notices that the worst has happened: the systems of the three operators are incompatible. So the city must reject connections built by two of the operators, and these are to be chosen so that the damage is minimized. What is the minimal number of offices which still can be connected to each other, possibly through intermediate offices, in the worst possible case.

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The city of Salavaara has six offices that need to be connected by a communication network. Each office must connect to all others, forming a complete graph with 15 connections total. Three operators compete for each connection, but their systems are incompatible. The city must choose one operator's network and reject the other two. We need to find the minimum number of offices that can stay connected in the worst possible scenario. The complete graph K6 has exactly 15 edges connecting all pairs of offices. These 15 edges must be partitioned among the three competing operators. Each edge is assigned to exactly one operator based on competition results. The city will later choose one operator and reject the other two. Our goal is to find the minimum connectivity guaranteed in the worst-case edge distribution. To solve this problem, we need to understand how the largest connected component depends on the number and structure of edges. With fewer edges, we get smaller components. With 5 edges as shown here, we can have a 4-cycle plus an isolated edge, giving us a largest component of size 4. The key insight is that we want to minimize the maximum component size across all three operators' subgraphs. The optimal strategy is to distribute the 15 edges equally: 5 edges per operator. We can construct a partition where each operator's subgraph has a largest connected component of exactly 4 vertices. This is achieved by giving each operator a 4-cycle plus one isolated edge. Since we've proven that no partition can guarantee all operators have LCC less than 4, this (5,5,5) distribution with maximum LCC of 4 is optimal. In conclusion, we have proven that the minimum number of offices that can remain connected in the worst possible case is 4. This is achieved when the adversary distributes the 15 edges among the three operators such that each operator's subgraph has a largest connected component of exactly 4 vertices. The city then chooses the best available option, guaranteeing connectivity for at least 4 offices. Therefore, the answer to this network optimization problem is 4.