Constructions of large planar networks with given degree and diameter

Fellows, Michael R.; Hell, Pavol; Seyffarth, Karen

doi:10.1002/(sici)1097-0037(199812)32:4<275::aid-net4>3.0.co;2-g

Cited by 10 publications

(14 citation statements)

References 2 publications

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“…As noted in [1], the diagram C k ∆ has maximum degree ∆ and readily satisfies the conditions of Proposition 2.3. Thus, the graph G(C k ∆ ) has maximum degree ∆ and diameter k. The edges ac and bg, however, are contained in the two closed walks of weight 2k + 2, namely abgeica and adgbxca.…”

Section: Large Planar Graphs With Odd Diametermentioning

confidence: 66%

“…Next we establish relations between D k ∆ and G(D k ∆ ); most of them already appeared in [1]. Lemma 2]).…”

Section: Multigraphs and Diagramsmentioning

confidence: 99%

“…Since β ≤ ⌊k/2⌋, for a thick pending edge e labelled by α(∆, β), the distance between any two vertices in G(D k ∆ ) belonging to the trees which replaced e is at most k as well. Condition (1) assures that any two vertices of G(D k ∆ ) belonging to pods replacing distinct thick edges are at distance at most k. Conditions (2) and (3) guarantee that the distance from a thin vertex to any vertex in a pod of G(D k ∆ ), and to any other thin vertex, is also at most k.…”

Section: Multigraphs and Diagramsmentioning

confidence: 99%

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Constructions of Large Graphs on Surfaces

Feria-Purón¹,

Pineda-Villavicencio²

2013

Graphs and Combinatorics

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We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface $\Sigma$ and integers $\Delta$ and $k$, determine the maximum order $N(\Delta,k,\Sigma)$ of a graph embeddable in $\Sigma$ with maximum degree $\Delta$ and diameter $k$. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known graphs. Given a surface $\Sigma$ of Euler genus $g$ and an odd diameter $k$, the current best asymptotic lower bound for $N(\Delta,k,\Sigma)$ is given by \[\sqrt{\frac{3}{8}g}\Delta^{\lfloor k/2\rfloor}.\] Our constructions produce new graphs of order \[\begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if $\Sigma$ is the Klein bottle}\\ $\frac{7}{2}+\sqrt{6g+\frac{1}{4}}$\Delta^{\lfloor k/2\rfloor}& \text{otherwise,}\end{cases}\] thus improving the former value by a factor of 4.Comment: 15 pages, 7 figure

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Section: Large Planar Graphs With Odd Diametermentioning

confidence: 66%

“…Next we establish relations between D k ∆ and G(D k ∆ ); most of them already appeared in [1]. Lemma 2]).…”

Section: Multigraphs and Diagramsmentioning

confidence: 99%

Section: Multigraphs and Diagramsmentioning

confidence: 99%

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Constructions of Large Graphs on Surfaces

Feria-Purón¹,

Pineda-Villavicencio²

2013

Graphs and Combinatorics

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“…We give two types of construction. With respect to distance vertex-colouring, similar constructions with the same leading constants 3/2 and 9/2 were given by Fellows, Hell and Seyffarth [10]. The constructions we give are simple and may furnish insight prior to our main result.…”

Section: Theorem 31 (Agnarsson and Halldórssonmentioning

confidence: 67%

“…Bounds in Proposition 1.2 for distance vertex-colouring were established generally for all t in [1] (upper bounds) and [10] (lower bounds). It is difficult to determine the precise values in Proposition 1.2, especially for distance vertex-colouring: lim sup d→∞ π 1 (d)/τ 1 (d) = 2 is the Four Colour Theorem [2,3], while lim sup d→∞ π 2 (d)/τ 2 (d) = 3/2 is the asymptotic confirmation of Wegner's Conjecture [12].…”

Section: Introductionmentioning

confidence: 99%

Tree-Like Distance Colouring for Planar Graphs of Sufficient Girth

Kang¹,

Loon²

2019

Electron. J. Combin.

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Given a multigraph G and a positive integer t, the distance-t chromatic index of G is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than t edges must receive different colours. Let π ′ t (d) and τ ′ t (d) be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree d. We have that π ′ t (d) is at most and at least a non-trivial constant multiple larger than τ ′ t (d). (We conjecture lim sup d→∞ π ′ 2 (d)/τ ′ 2 (d) = 9/4 in particular.) We prove for odd t the existence of a quantity g depending only on t such that the distance-t chromatic index of any planar multigraph of maximum degree d and girth at least g is at most τ ′ t (d) if d is sufficiently large. Such a quantity does not exist for even t. We also show a related, similar phenomenon for distance vertex-colouring.

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Maximum Order of Planar Digraphs

Simanjuntak

Miller

2005

Combinatorial Geometry and Graph Theory

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Constructions of large planar networks with given degree and diameter

Cited by 10 publications

References 2 publications

Constructions of Large Graphs on Surfaces

Constructions of Large Graphs on Surfaces

Tree-Like Distance Colouring for Planar Graphs of Sufficient Girth

Maximum Order of Planar Digraphs

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