2022 users, friendships. New friendship only if 2 common friends. Min friendships needed for everyone to become friends?
The minimum number of initial friendships required is 2020. For any graph with vertices to eventually become fully connected, it must initially be connected. The minimum number of edges in a connected graph with 2021 vertices is . A configuration of 2020 edges (such as a double-star where two adjacent vertices are connected to everyone else) is sufficient to propagate to a complete graph.
The minimum number of initial friendships required is 2023. For any graph with vertices to eventually become fully connected, it must initially be connected. The minimum number of edges in a connected graph with 2024 vertices is . A configuration of 2023 edges (such as a double-star where two adjacent vertices are connected to everyone else) is sufficient to propagate to a complete graph.
The minimum number of initial friendships required is 2021. For any graph with vertices to eventually become fully connected, it must initially be connected. The minimum number of edges in a connected graph with 2022 vertices is . A configuration of 2021 edges (such as a double-star where two adjacent vertices are connected to everyone else) is sufficient to propagate to a complete graph.
The minimum number of initial friendships required is 2022. For any graph with vertices to eventually become fully connected, it must initially be connected. The minimum number of edges in a connected graph with 2023 vertices is . A configuration of 2022 edges (such as a double-star where two adjacent vertices are connected to everyone else) is sufficient to propagate to a complete graph.
Hint 1: Note that the propagation rule only adds edges within existing connected components.
Hint 2: What is the minimum number of edges needed to connect a graph of 2022 vertices? (Recall that a connected graph has at least edges).
Hint 3: Construct an initial star or double-star configuration with 2021 edges and show that it eventually propagates to a complete graph.
Step 1 (Graph formulation): Let the social network be a graph with 2022 vertices. A new edge (friendship) can be added between two vertices if they share at least two common neighbors. We want to find the minimum number of initial edges required to eventually connect all vertices.
Step 2 (Lower Bound): Any graph of vertices requires at least edges to be connected. For , at least edges are required. If the graph has fewer than 2021 edges, it is disconnected, and since the edge propagation rule only adds edges between vertices in the same connected component, the graph can never become fully connected.
Step 3 (Sufficient Construction): We can achieve full connectivity with exactly 2021 edges by constructing a tree-like cycle. Specifically, we arrange the vertices in a line and connect them in a cycle or a double-star configuration where two central vertices are connected to all other vertices. This initial configuration allows edges to propagate rapidly, eventually forming a complete graph . Thus, the minimum number of initial friendships is 2021.
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