Largest planar graphs of diameter two and fixed maximum degree

Hell, Pavol; Seyffarth, Karen

doi:10.1016/0012-365x(93)90166-q

Cited by 29 publications

(20 citation statements)

References 3 publications

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“…We now consider the degree-diameter problem for graphs with given Euler genus. Note that the case of planar graphs has been widely studied [12,13,19,25,30,31].Šiagiová and Simanjuntak [27] proved that for every graph G with Euler genus g, Theorem 14. For all > 0 there is a constant c such that for every graph G with Euler genus g, maximum degree ∆ and diameter k,…”

Section: Now Consider the Even K Case For ∆mentioning

confidence: 99%

The Degree-Diameter Problem for Sparse Graph Classes

Pineda-Villavicencio

Wood

2015

Electron. J. Combin.

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A. The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree ∆ and diameter k. For fixed k, the answer is Θ(∆ k ).We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(∆ k−1 ), and for graphs of bounded arboricity the answer is Θ(∆ k/2 ), in both cases for fixed k. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs.

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Section: Now Consider the Even K Case For ∆mentioning

confidence: 99%

The Degree-Diameter Problem for Sparse Graph Classes

Pineda-Villavicencio

Wood

2015

Electron. J. Combin.

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“…Consider the graph G in Figure 1; this graph was constructed in [2] to show that Let S be a surface other than the sphere (for the ''spherical''result we refer to [5]) and let G be as in Figure 1 with vertices a 1 , . . .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…One of the possible ways was proposed in [2] where it is proved that a planar graph of diameter two and maximum degree d ≥ 8 has at most (3/2)d + 1 vertices. In addition, as shown in [5], this bound is best possible in the following (strong) sense: For each d ≥ 8 there exists a planar triangulation of diameter two and maximum degree d that contains exactly (3/2)d + 1 vertices.…”

Section: Introductionmentioning

confidence: 98%

Extremal graphs of diameter two and given maximum degree, embeddable in a fixed surface

Knor

?ir�?

1997

J. Graph Theory

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It is known that for each d there exists a graph of diameter two and maximum degree d which has at least ⌈(d/2)⌉ ⌈(d + 2)/2⌉ vertices. In contrast with this, we prove that for every surface S there is a constant ds such that each graph of diameter two and maximum degree d ≥ ds, which is embeddable in S, has at most ⌊(3/2)d⌋ + 1 vertices. Moreover, this upper bound is best possible, and we show that extremal graphs can be found among surface triangulations. © 1997 John Wiley & Sons, Inc.

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“…She proved that, in this case, the number of vertices is n 3 2 ∆ + 1 if ∆ 8, and that this bound is best possible. Later, Hell and Seyffarth [9] showed that this result also holds for the larger class of all simple planar graphs. Yang, Lin, and Dai [19] solved the remaining case ∆ < 8 with D = 2, for both graph classes.…”

Section: Introductionmentioning

confidence: 88%

The Degree/Diameter Problem in Maximal Planar Bipartite graphs

Dalfó

Huemer

Salas

2016

Electron. J. Combin.

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The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

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Largest planar graphs of diameter two and fixed maximum degree

Cited by 29 publications

References 3 publications

The Degree-Diameter Problem for Sparse Graph Classes

The Degree-Diameter Problem for Sparse Graph Classes

Extremal graphs of diameter two and given maximum degree, embeddable in a fixed surface

The Degree/Diameter Problem in Maximal Planar Bipartite graphs

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