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The Degree-Diameter Problem for Outerplanar Graphs

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Dankelmann P., Jonck E., Vetrík T.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 3

823-834

For positive integers Δ and D we define nΔ,D to be the largest number of vertices in an outerplanar graph of given maximum degree Δ and diameter D. We prove that [...] nΔ,D=ΔD2+O (ΔD2−1) $n_{\Delta ,D} = \Delta ^{{D \over 2}} + O\left( {\Delta ^{{D \over 2} - 1} } \right)$ is even, and [...] nΔ,D=3ΔD−12+O (ΔD−12−1) $n_{\Delta ,D} = 3\Delta ^{{{D - 1} \over 2}} + O\left( {\Delta ^{{{D - 1} \over 2} - 1} } \right)$ if D is odd. We then extend our result to maximal outerplanar graphs by showing that the maximum number of vertices in a maximal outerplanar graph of maximum degree Δ and diameter D asymptotically equals nΔ,D.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Dankelmann P.

1 Jonck E.

1 Vetrík T.

1 2017

Wyniki wyszukiwania

The Degree-Diameter Problem for Outerplanar Graphs