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

Kierstead, H. A.; Kostochka, Alexandr; Yu, Gexin

doi:10.1017/cbo9781107325975.006

Cited by 22 publications

(24 citation statements)

References 72 publications

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“…Theorem 1 improved previous bounds by Alon and Yuster [1], who showed that δ(H, n) ≤ (1 − 1/χ(H))n + o(n), and by Komlós, Sárközy and Szemerédi [11], who replaced the o(n)-term by a constant depending only on H. Further related results are discussed in the surveys [7,8,12,15,21].…”

Section: Introductionsupporting

confidence: 77%

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An Ore-type Theorem for Perfect Packings in Graphs

Kühn

Osthus²,

Treglown³

2009

SIAM J. Discrete Math.

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We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let δOre(H, n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x) + d(y) ≥ k for all non-adjacent x = y ∈ V (G) contains a perfect H-packing. We determine limn→∞ δOre(H, n)/n.

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Section: Introductionsupporting

confidence: 77%

“…for the Pósa-Seymour conjecture which states that every graph G on n vertices with δ(G) ≥ r r+1 n contains the rth power of a Hamilton cycle. [7] contains a discussion of other Ore-type results.…”

Section: 5mentioning

confidence: 99%

An Ore-type Theorem for Perfect Packings in Graphs

Kühn

Osthus²,

Treglown³

2009

SIAM J. Discrete Math.

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“…His theorem was generalized by Ore [24]: Assume that n ≥ 3, G is a graph of order n and for every xy ∈ E(G) we have deg(x) + deg(y) ≥ n. Then G has a Hamilton cycle. This result motivates the notion of the Ore-degree of an edge [17,21]: the Ore-degree of xy is the sum θ(x, y) = deg(x) + deg(y) Claim 3.3. Assume that H is a graph with θ(H) = 4.…”

Section: Introductionmentioning

confidence: 76%

Embedding Graphs Having Ore-Degree at Most Five

Csaba¹,

Nagy-György²

2019

SIAM J. Discrete Math.

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Let H and G be graphs on n vertices, where n is sufficiently large. We prove that if H has Ore-degree at most 5 and G has minimum degree at least 2n/3 then H is a subgraph of G.of the degrees of its endpoints. The Ore-degree of G, denoted by θ(G), is the maximum Ore-degree in E(G). The embedding problems that include minimum and maximum degree conditions are called Dirac-type, while those involving the Ore-degree are called Ore-type problems in the literature.An excellent source of embedding results and conjectures is the survey [17] by Kierstead, Kostochka and Yu, in which several Ore-type questions are considered as well. One can easily formulate an Ore-type problem by replacing the maximum degree in a Dirac-type embedding problem by half of the Ore-degree of the graph, or if one considers a packing version, then one can replace even both maximum degrees. In some cases the resulting questions were solved. For example, Kostochka and Yu proved [22] that if G 1 and G 2 are graphs of order n such that θ(G 1 )∆(G 2 ) < n then G 1 and G 2 pack. This is in fact a half-Ore version of the famous packing result of Sauer and Spencer [25]: if ∆(G 1 ) · ∆(G 2 ) < n/2 then G 1 and G 2 pack. We remark that the full-Ore version of the Sauer-Spencer theorem, when both maximum degrees are replaced by half of the corresponding Ore-degree, is open. Another important Dirac-type theorem for which the half-Ore version was proved is by Aigner-Brandt [1] and Alon-Fischer [2] on embedding 2-factors. The half-Ore version was proved by Kostochka and Yu [23].

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“…The main open conjecture on the subject is the one of Bollobás and Eldridge [3] asserting that if (∆(G) + 1)(∆(H) + 1) ≤ n + 1 then G and H pack. Sauer and Spencer ( [9], see also Catlin's paper [4]) proved that this is the case if 2∆(G)∆(H) < n. For a survey of packing results including many extensions, variants and relevant references, see [7].…”

Section: Discussionmentioning

confidence: 99%

The Turán number of sparse spanning graphs

Alon

Yuster

2013

Journal of Combinatorial Theory, Series B

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For a graph H, the extremal number ex(n, H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H. Let δ(H) > 0 and ∆(H) denote the minimum degree and maximum degree of H, respectively. We prove that for all n sufficiently large, if H is any graph of order n with ∆(H) ≤ √ n/200, then ex(n, H) = n−1 2 +δ(H)−1. The condition on the maximum degree is tight up to a constant factor. This generalizes a classical result of Ore for the case H = C n , and resolves, in a strong form, a conjecture of Glebov, Person, and Weps for the case of graphs. A counter-example to their more general conjecture concerning the extremal number of bounded degree spanning hypergraphs is also given.

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

Abstract: We discuss recent progress and unsolved problems concerning extremal graph packing, emphasizing connections between Dirac-type and Ore-type problems. Extra attention is paid to coloring, and especially equitable coloring, of graphs.

Cited by 22 publications

References 72 publications

An Ore-type Theorem for Perfect Packings in Graphs

An Ore-type Theorem for Perfect Packings in Graphs

Embedding Graphs Having Ore-Degree at Most Five

The Turán number of sparse spanning graphs

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