This task has not been reviewed, its validity can be dubious.
Platonic solids
Task number: 4251
Consider a connected \( k \)-regular planar graph \( G \) on \(n\) vertices, \( k \geq 3 \), which allow a planar drawing where all faces have degree \( l \).
Variant
Show that: \(n(2k + 2l -kl) = 4l\).
Solution
We use the fact \( \sum \limits_{v \in V (G)} \deg v = 2m = \sum \limits_ {f \in F} \deg f, \) where \( m \) is the number of edges and \( F \) denotes the set of all faces.
Variant
Derive that the only options for \( (k, l) \) are \( (3{,}3) \), \( (3{,}4) \), \( ( 3{,}5) \), \( (4{,}3) \) and \( (5{,}3) \). In each of the variants, specify the number of vertices and edges of the corresponding graph.
Variant
For each of the options in the previous point, find a graph that satisfies the assignment.
