Large planar graphs with given diameter and maximum degree

AbstractWe consider the problem of determining the maximum number of vertices in a planar graph with given maximum degree Δ and diameter k. This number has previously been exactly determined when k = 2. We show here that when k = 3, the number is roughly between 4.5Δ and 8Δ. We also show that in general the number is Θ(Δ⌊k2⌋) for any fixed value of k