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**1 - 1**of**1**### New Bounds for Largest Planar Graphs With Fixed Maximum Degree and Diameter

"... Let p(\Delta; D) be the largest number of vertices in a planar graph with maximum degree \Delta and diameter D. Let pr(\Delta; D) be the largest number of vertices in a planar \Delta-regular graph with diameter D. We improve many of the best-known upper bounds for p(\Delta; D) and pr(\Delta; D) ..."

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Let p(\Delta; D) be the largest number of vertices in a planar graph with maximum degree \Delta and diameter D. Let pr(\Delta; D) be the largest number of vertices in a planar \Delta-regular graph with diameter D. We improve many of the best-known upper bounds for p(\Delta; D) and pr(\Delta; D) and most of the lower bounds for pr(\Delta; D). In particular, we prove that p(3; D) 2 D+1 \Gamma 1, p(4; D) (5 + 23 \Delta 3 D\Gamma3 )=2, and p(5; D) (8 + 46 \Delta 4 D\Gamma2 )=3. 1 Figure 1: The planar 3-regular graph with diameter 2 and 6 vertices 1 Preliminaries Lately, there has been considerable progress and interest in the degreediameter problem: to find the largest graph with specified maximum degree and diameter [1, 2, 3, 4, 6, 8, 10]. We consider the same problem restricted to planar graphs. Let p(\Delta; D) be the largest number of vertices in a planar graph with maximum degree \Delta and diameter D. Let pr(\Delta; D) be the largest number of vertices in a planar ...