Packing d-degenerate graphs

Bollobás, Béla; Kostochka, Alexandr; Nakprasit, Kittikorn

doi:10.1016/j.jctb.2007.05.002

Cited by 17 publications

(16 citation statements)

References 9 publications

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“…This conjecture has been proved in the case ∆ 1 ≤ 2 by Aigner and Brandt [1] and Alon and Fisher (for sufficiently large n) [2], and in the case when ∆ 1 = 3 and n is huge by Csaba, Shokoufandeh, and Szemerédi [8]. Bollobás, Kostochka and Nakprasit [5] proved a strengthening of the conjecture when G 1 is d-degenerate and d < ∆ 1 /40. Eaton [9] showed that under the given condition, there is a near-packing of degree 1 of G 1 and G 2 , that is, an embedding of the two graphs into a common vertex set such that the maximum degree of the subgraph defined by the edges common to both copies is 1.…”

Section: Introductionmentioning

confidence: 88%

On a graph packing conjecture by Bollobás, Eldridge and Catlin

Kaul

Kostochka

2008

Combinatorica

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Two graphs G1 and G2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (∆(G1) + 1)(∆(G2) + 1) ≤ n + 1, then G1 and G2 pack. Towards this conjecture, we show that for ∆(G1), ∆(G2) ≥ 300, if (∆(G1)+1)(∆(G2)+1) ≤ 0.6n+1, then G1 and G2 pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G1 and G2 pack if ∆(G1)∆(G2) < 0.5n.

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Section: Introductionmentioning

confidence: 88%

On a graph packing conjecture by Bollobás, Eldridge and Catlin

Kaul

Kostochka

2008

Combinatorica

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“…Aigner and Brandt [1] and independently (for huge n) Alon and Fisher [2] settled the conjecture in the case ∆ 1 ≤ 2 (this particular case was conjectured by Sauer and Spencer [68]). Csaba, Shokoufandeh, and Szemerédi [21] proved the BECconjecture for ∆ 1 = 3 and huge n. Bollobás, Kostochka and Nakprasit [12] showed that although the BEC-conjecture is sharp, if one of the two graphs is sparse, to be precise, d-degenerate for a small d, then much weaker conditions on ∆ 1 and ∆ 2 imply the existence of a packing. Recall that a graph G is d-degenerate if every subgraph G of G has a vertex of degree (in G ) at most d. In this case, the vertices of G can be ordered so that each vertex has fewer than col(G) := d + 1 neighbors that precede it.…”

Section: Theorem 24mentioning

confidence: 99%

“…Recall that a graph G is d-degenerate if every subgraph G of G has a vertex of degree (in G ) at most d. In this case, the vertices of G can be ordered so that each vertex has fewer than col(G) := d + 1 neighbors that precede it. Theorem 2.10 (Bollobás, Kostochka and Nakprasit [12]) Let d ≥ 2. Let G 1 be a d-degenerate graph of order n and maximum degree ∆ 1 and G 2 be a graph of order n and maximum degree at most ∆ 2 .…”

Section: Theorem 24mentioning

confidence: 99%

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Extremal graph packing problems: Ore-type versus Dirac-type

Kierstead¹,

Kostochka²,

Yu³

2009

Surveys in Combinatorics 2009

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“…al use the method of random separation to obtain fixedparameter tractable algorithms for the problem of finding induced cycles and trees in degenerate graphs [5]. Degenerate graphs have also been studied from a theoretical point of view [6,14,4].…”

Section: Introductionmentioning

confidence: 99%

Enumerating and Generating Labeled k-degenerate Graphs

Bauer

Krug

Wagner

2010

2010 Proceedings of the Seventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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A k-degenerate graph is a graph in which every induced subgraph has a vertex with degree at most k. The class of k-degenerate graphs is interesting from a theoretical point of view and it plays an interesting role in the theory of fixed parameter tractability since some otherwise W[2]-hard domination problems become fixed-parameter tractable for k-degenerate graphs.It is a well-known fact that the k-degenerate graphs are exactly the graphs whose vertex-set can be well-ordered such that each vertex is incident to at most k larger vertices with respect to this ordering. A well-ordered k-degenerate graph is a labeled graph with vertex-labels 1, . . . , n such that the ordering of the vertices by their labels is a well-ordering of the graph.We consider the problem of enumerating and generating well-ordered k-degenerate graphs with a given number of vertices and with a given number of vertices and edges, respectively, uniformly at random. By generating wellordered k-degenerate graphs we generate at least one labeled copy of each unlabeled k-degenerate graph and we filter some but not all isomorphies compared to the classical labeled approach.We also introduce the class of strongly k-degenerate graphs, which are k-degenerate graphs with minimum degree k. These graphs are a natural generalization of k-regular graphs which can be used in order to generate graphs with predefined core-decomposition.We present efficient algorithms for generating wellordered k-degenerate graphs with given number of vertices (and edges). After a precomputation which must only be performed once when generating more than one well-ordered k-degenerate graph these algorithms are almost optimal. Additionally, we present complete non-uniform generators for these classes with optimal running time. We also present an efficient and complete generator for well-ordered strongly k-degenerate graphs with given number of vertices (and edges). Finally, we present efficient algorithms for enumerating well-ordered k-degenerate and strongly k-degenerate graphs.

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Packing d-degenerate graphs

Cited by 17 publications

References 9 publications

On a graph packing conjecture by Bollobás, Eldridge and Catlin

On a graph packing conjecture by Bollobás, Eldridge and Catlin

Extremal graph packing problems: Ore-type versus Dirac-type

Enumerating and Generating Labeled k-degenerate Graphs

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