Given is a tree $G$ with $2023$ vertices. The longest path in the graph has length $2n$. A vertex is called good if it has degree at most $6$. Find the smallest possible value of $n$ if there doesn't exist a vertex having $6$ good neighbors.
Source: Bulgarian Spring Tournament 2023 11.4
Tags: combinatorics
Given is a tree $G$ with $2023$ vertices. The longest path in the graph has length $2n$. A vertex is called good if it has degree at most $6$. Find the smallest possible value of $n$ if there doesn't exist a vertex having $6$ good neighbors.