Cubic graphs

Greenlaw, Raymond; Petreschi, Rossella

doi:10.1145/234782.234783

Cited by 44 publications

(22 citation statements)

References 34 publications

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“…A survey of cubic graphs is well described by Greenlaw (1995). In this paper we have studied edge coloring, which is a well-known subfield of graph theory, and presented algorithms seeking a kind of generalized 3-edge colorings known as Fan-Raspaud colorings, which can represent a mathematical model of pair assignments for parallel computations in cubic computer networks.…”

Section: Resultsmentioning

confidence: 99%

The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

Formanowicz

Tanaź

2012

International Journal of Applied Mathematics and Computer Science

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It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan-Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan-Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another possible application of the algorithms is that of being a tool for mathematicians working in the field of cubic graph theory, for discovering edge colorings with certain mathematical properties and formulating new conjectures related to the Fan-Raspaud conjecture.

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Section: Resultsmentioning

confidence: 99%

The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

Formanowicz

Tanaź

2012

International Journal of Applied Mathematics and Computer Science

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“…Moreover, GЈ is a partial graph of a grid; hence, it is planar and bipartite. ■ Moreover, we reobtain a well-known NP-Completeness result (see, for instance, [13]): Proof. Take the 2-section graph of GЈ in the above corollary.…”

Section: Fig 2 Representation Of Variablesmentioning

confidence: 94%

Cardinality constrained path covering problems in grid graphs

2004

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In this article we continue our study on the complexity of Path Covering Problems started in [2]. Here, taking one further step, we investigate the complexity of the problem on grids. For special classes of grids (general grids, grids with a fixed number of rows, ladders), and several special unweighted path collections (general paths, paths of length 2, L-shaped paths, pipes, hooks, staples) we either give polynomial-time algorithms or prove NPcompleteness results.

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“…Moreover, in a certain sense cubic graphs are the generic regular graphs (see e.g. Bollobás [2] or Greenlaw and Petreschi [3]). 1 So let us consider the set of all regular cubic graphs with n vertices.…”

Section: A Numerical Experimentsmentioning

confidence: 99%

Clustering of spectra and fractals of regular graphs

Ejov

Filar

Lucas

et al. 2007

Journal of Mathematical Analysis and Applications

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We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labeled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara-Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind. A numerical experimentFor the sake of simplicity we will pay our attention mainly to cubic graphs (or, in other words, to regular graphs of degree d = 3). This assumption is not restrictive since all our considerations remain valid for regular graphs of degree d > 3 with obvious minor modifications. Moreover, in

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Cubic graphs

Cited by 44 publications

References 34 publications

The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

Cardinality constrained path covering problems in grid graphs

Clustering of spectra and fractals of regular graphs

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