Maximal cubic graphs with diameter 4

Buset, Dominique

doi:10.1016/s0166-218x(99)00204-8

Cited by 7 publications

(4 citation statements)

References 13 publications

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“…For maximum degree 3 and diameter 4 the Moore bound is M 3,4 = 46. Jørgensen [216] and [217] proved that there is no (3, 4)-graph of defect 2 or 4, and Buset [78] showed that there is no (3, 4)-graph of defect 6. Therefore, the two known non-isomorphic (3, 4)graphs of defect 8 constructed by Doty [131] and by von Conta [110] For maximum degree 6 and diameter 2 the Moore bound is M 6,2 = 37.…”

Section: Tables Of Large Graphsmentioning

confidence: 99%

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Miller¹,

Širáň‬²

2013

Electron. J. Combin.

220

224

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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter.General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'.This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

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Section: Tables Of Large Graphsmentioning

confidence: 99%

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Miller¹,

Širáň‬²

2013

Electron. J. Combin.

220

224

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“…By mapping the cycle D 2 to C, the vertex q 2 to x, the vertex q 3 to x ′ , the cycle D 1 to C 1 , the cycle D 3 to C 2 , the vertex q 1 to y and the vertex q 4 to y ′ , we obtain that q 1 and q 4 are repeat vertices in the repeat cycle of D 2 . Therefore, since q 4 ∈ D 4 , it follows that D 2 and D 4 are repeat cycles and q 1 = q 5 . As a consequence, there is in Γ a cycle q 1 p 1 p 3 p 5 q 5 of length Suppose that Γ contains a subgraph Θ isomorphic to Θ D , with branch vertices a and b.…”

Section: Repeats Of Cyclesmentioning

confidence: 99%

“…Only a few values of N(∆, D) are known at present. With the exception of N(4, 2) = M(4, 2) − 2 (see [3]), N(5, 2) = M(5, 2) − 2 (see [20]), N(6, 2) = M(6, 2) − 5 (see [19]), N(3, 3) = M(3, 3) − 2 (see [14]) and N(3, 4) = M(3, 4) − 8 (see [4]), the other known values of N(∆, D) are those for which there exists a Moore graph.…”

Section: Introductionmentioning

confidence: 99%

“…An upper bound for N(∆, D) is given by the Moore bound M(∆, D),Graphs of degree ∆, diameter D and order M(∆, D) are called Moore graphs.Only a few values of N(∆, D) are known at present. With the exception of N(4, 2) = M(4, 2) − 2 (see [3]), N(5, 2) = M(5, 2) − 2 (see [20]), N(6, 2) = M(6, 2) − 5 (see [19]), N(3, 3) = M(3, 3) − 2 (see [14]) and N(3, 4) = M(3, 4) − 8 (see [4]), the other known values of N(∆, D) are those for which there exists a Moore graph.Moore graphs are very rare. For ∆ = 2 and D ≥ 1 they are the cycles on 2D + 1 vertices, whereas for D = 1 and ∆ ≥ 2 they are the complete graphs on ∆ + 1 vertices.…”

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confidence: 99%

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On graphs of defect at most 2

Feria-Purón

Miller

Pineda-Villavicencio

2011

Discrete Applied Mathematics

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In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the Moore bound M({\Delta},D) represents an upper bound for N({\Delta},D). Graphs of maximum degree {\Delta}, diameter D and order M({\Delta},D), called Moore graphs, turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree {\Delta} \geq 2, diameter D \geq 1 and order M({\Delta},D) - {\epsilon} with small {\epsilon} > 0, that is, ({\Delta},D,-{\epsilon})-graphs. The parameter {\epsilon} is called the defect. Graphs of defect 1 exist only for {\Delta} = 2. When {\epsilon} > 1, ({\Delta},D,-{\epsilon})-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a ({\Delta},D,-2)-graph with {\Delta} \geq 4 and D \geq 4 is 2D. Second, and most important, we prove the non-existence of ({\Delta},D,-2)-graphs with even {\Delta} \geq 4 and D \geq 4; this outcome, together with a proof on the non-existence of (4, 3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2. Such a catalogue is only the second census of ({\Delta},D,-2)-graphs known at present, the first being the one of (3,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2 [14]. Other results of this paper include necessary conditions for the existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 4, and the non-existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 5 such that {\Delta} \equiv 0, 2 (mod D).Comment: 22 pages, 11 Postscript figure

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New largest known graphs of diameter 6

Pineda-Villavicencio

Gómez²,

Miller

et al. 2008

Networks

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In the pursuit of obtaining largest graphs of given maximum degree and diameter D, many construction techniques have been developed. Compounding of graphs is one such technique. In this article, by means of the compounding of complete graphs into a bipartite Moore graph of diameter 6, we obtain a family of large graphs of the same diameter. For maximum degrees = 5, 6, 9, 12, and 14, members of this family constitute the largest known graphs of diameter 6.

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Maximal cubic graphs with diameter 4

Cited by 7 publications

References 13 publications

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

On graphs of defect at most 2

New largest known graphs of diameter 6

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