A large class is to be divided into teams and each student must be a member of exactly one team. However, each student dislikes three of their classmates. Dislike between students need not be mutual. If the teams do not need to be equally sized, how many must be created so that no student is the teammate of someone they dislike?

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∻Solution by Steps∻ ‖ step 1 ⋮ Determine the minimum number of colors (teams) needed for graph coloring. ‖ ‖ step 2 ⋮ The given graph is a regular graph where each vertex is connected to 3 other vertices. ‖ ‖ step 3 ⋮ Use the concept of graph coloring where no two adjacent vertices (students who dislike each other) can have the same color (be on the same team). ‖ ‖ step 4 ⋮ Apply Brooks' Theorem, which states that for a connected graph with maximum degree Δ, the chromatic number is at most Δ unless the graph is a complete graph or an odd cycle, which require Δ + 1 colors. ‖ ‖ step 5 ⋮ Since each student dislikes exactly 3 others, the maximum degree Δ is 3. The graph is not a complete graph or an odd cycle. ‖ ‖ step 6 ⋮ Therefore, the chromatic number, which is the minimum number of colors (teams), is at most 3. ‖ ∻Answer∻ ⚹ At most 3 teams are needed to ensure no student is the teammate of someone they dislike. ⚹ ∻Key Concept∻ ⚹Graph Coloring and Brooks' Theorem⚹ ∻Explanation∻ ⚹The problem is analogous to a graph coloring problem where each vertex represents a student and each edge represents mutual dislike. The minimum number of colors (teams) needed so that no adjacent vertices (students who dislike each other) share the same color (team) is determined by Brooks' Theorem.⚹◊What is the relationship between the number of students, the number of disliked classmates, and the minimum number of teams needed for no student to be paired with someone they dislike?⍭ Generate me a similar question◊

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