Visitor in 3-regular planar graph park turns left/right alternatingly. Max times entering any vertex?
The maximum number of times the visitor can enter any single vertex is 5. Since the graph is 5-regular, each vertex has exactly 5 incident edges, meaning a visitor entering and exiting must use distinct pairs of edges. Through face-coloring or topological invariants of planar graphs, it can be proven that a vertex cannot be visited more than 5 times without violating the alternating left-right turn sequence.
The maximum number of times the visitor can enter any single vertex is 2. Since the graph is 2-regular, each vertex has exactly 2 incident edges, meaning a visitor entering and exiting must use distinct pairs of edges. Through face-coloring or topological invariants of planar graphs, it can be proven that a vertex cannot be visited more than 2 times without violating the alternating left-right turn sequence.
The maximum number of times the visitor can enter any single vertex is 3. Since the graph is 3-regular, each vertex has exactly 3 incident edges, meaning a visitor entering and exiting must use distinct pairs of edges. Through face-coloring or topological invariants of planar graphs, it can be proven that a vertex cannot be visited more than 3 times without violating the alternating left-right turn sequence.
The maximum number of times the visitor can enter any single vertex is 4. Since the graph is 4-regular, each vertex has exactly 4 incident edges, meaning a visitor entering and exiting must use distinct pairs of edges. Through face-coloring or topological invariants of planar graphs, it can be proven that a vertex cannot be visited more than 4 times without violating the alternating left-right turn sequence.
Hint 1: Consider the degree of each vertex in a 3-regular graph. Each entry and exit uses 2 distinct edges.
Hint 2: Since each vertex has degree 3, a visitor can enter a vertex at most twice before having only one unused edge remaining.
Hint 3: Show that the visitor can enter a vertex a third time but is then forced to stop or backtrack, proving 3 is the maximum.
Step 1 (Alternating Turn Rule): A visitor in a 3-regular planar graph park walks along the edges, turning left and right alternatingly. We want to find the maximum number of times the visitor can enter any single vertex .
Step 2 (Degree Constraints): Since the graph is 3-regular, each vertex has exactly 3 incident edges. A visitor entering a vertex must exit along a different edge. An entry and exit uses exactly 2 of the 3 incident edges. Therefore, a visitor can enter and exit a vertex at most once using any pair of edges.
Step 3 (Maximizing Entries): To enter a vertex a third time, the visitor must use the remaining edge. However, they cannot exit without reusing an edge. Under the alternating left-right turn rule in a planar graph, reusing an edge is impossible without creating an invalid loop. By setting up a face-marking invariant and using Euler's formula, we prove that the visitor can enter any vertex at most 3 times.
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