The Degree-Diameter Problem for Sparse Graph Classes

Pineda-Villavicencio, Guillermo; Wood, David R.

doi:10.37236/4313

Cited by 4 publications

(11 citation statements)

References 24 publications

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“…Our results and those from [10] imply that, for a fixed odd diameter k, N(∆, k, Σ) is asymptotically larger than N(∆, k, P). For even diameter, however, we believe this is not the case; thus, we dare to conjecture the following.…”

Section: Discussionsupporting

confidence: 61%

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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: Discussionsupporting

confidence: 61%

“…For odd k we think the actual assymptotic value of N(∆, k, Σ) is (c 1 + c 2 √ g)∆ ⌊k/2⌋ , where c 1 and c 2 are absolute constants. The case of g = 0 was proved in [10].…”

Section: Discussionmentioning

confidence: 99%

Constructions of Large Graphs on Surfaces

Feria-Purón¹,

Pineda-Villavicencio²

2013

Graphs and Combinatorics

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“…A generalisation to the class G H of H-minor-free graphs, with H a fixed graph, was studied in [15]. The current best upper bound of N(∆, k, G H ) ≤ 4k(c|H| log |H|) k ∆ ⌊k/2⌋ was given in [15,Sec. 4].…”

Section: Conjecturementioning

confidence: 99%

On the maximum order of graphs embedded in surfaces

Nevo

Pineda-Villavicencio

Wood

2016

Journal of Combinatorial Theory, Series B

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The maximum number of vertices in a graph of maximum degree Δ ≥ 3 and fixed diameter k ≥ 2 is upper bounded by (1 + o(1))(Δ − 1) k . If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of (2 + o(1))(Δ − 1) k/2 for a fixed k. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus g behave like trees, in the sense that, for large Δ, such graphs have orders bounded from above bywhere c is an absolute constant. This result represents a qualitative improvement over all previous results, even for planar graphs of odd diameter k. With respect to lower bounds, we construct graphs of Euler genus g, odd diameter k, and order c( √ g + 1)(Δ − 1) k/2 for some absolute constant c > 0. Our results answer in the negative a question of Miller and Širáň (2005).

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“…For planar graphs with even diameter D = 2k and maximum degree ∆, Tishchenko [17] obtained the lower bound 3∆ 2 (∆−1) k −1 ∆−2 + 1, and proved that this is also an upper bound for cases with large ∆, concretely for ∆ 6(12 k + 1 ). Related bounds on the (∆, D) problem for sparse graph classes and for graphs embedded on surfaces can be found in Nevo, Pineda-Villavicencio and Wood [11], and Pineda-Villavicencio and Wood [12]. In particular, in [12] the following upper bounds on the number n of vertices in a graph with diameter D and maximum degree ∆ are proved: • Graphs with arboricity b: n 4D(2b) D ∆ D/2 + 1.…”

Section: Introductionmentioning

confidence: 94%

“…Related bounds on the (∆, D) problem for sparse graph classes and for graphs embedded on surfaces can be found in Nevo, Pineda-Villavicencio and Wood [11], and Pineda-Villavicencio and Wood [12]. In particular, in [12] the following upper bounds on the number n of vertices in a graph with diameter D and maximum degree ∆ are proved: • Graphs with arboricity b: n 4D(2b) D ∆ D/2 + 1.…”

Section: Introductionmentioning

confidence: 94%

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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The Degree-Diameter Problem for Sparse Graph Classes

Cited by 4 publications

References 24 publications

Constructions of Large Graphs on Surfaces

Constructions of Large Graphs on Surfaces

On the maximum order of graphs embedded in surfaces

The Degree/Diameter Problem in Maximal Planar Bipartite graphs

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