Best Monotone Degree Conditions for Graph Properties: A Survey

Bauer, Douglas C.; Broersma, Hajo; Heuvel, Jan van den; Kahl, Nathan; Nevo, A.; Schmeichel, E. F.; Woodall, Douglas R.; Yatauro, M.

doi:10.1007/s00373-014-1465-6

Cited by 22 publications

(13 citation statements)

References 45 publications

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“…The Chvátal condition is a monotone best possible degree condition for hamiltonicity , and is a thorough survey of monotone best possible degree criteria for a number of graph properties. However, it is easy to show that neither the Chvátal condition or the weak Chvátal condition is monotone best possible for the property

m (G, 2) = 2

.…”

Section: Resultsmentioning

confidence: 99%

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Dairyko

Ferrara

Lidický

et al. 2019

Journal of Graph Theory

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Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph G begin in one of two states, “dormant” or “active.” Given a fixed positive integer r, a dormant vertex becomes active if at any stage it has at least r active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices A, we say that G r‐percolates (from A) if every vertex in G becomes active after some number of steps. Let m(G,r) denote the minimum size of a set A such that G r‐percolates from A. Bootstrap percolation has been studied in a number of settings and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree‐based density conditions that ensure m(G,2)=2. In particular, we give an Ore‐type degree sum result that states that if a graph G satisfies σ2(G)≥n−2, then either m(G,2)=2 or G is in one of a small number of classes of exceptional graphs. (Here, σ2(G) is the minimum sum of degrees of two nonadjacent vertices in G.) We also give a Chvátal‐type degree condition: If G is a graph with degree sequence d1≤d2≤⋯≤dn such that di≥i+1 or dn−i≥n−i−1 for all 1≤i

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m (G, 2) = 2

.…”

Section: Resultsmentioning

confidence: 99%

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Dairyko

Ferrara

Lidický

et al. 2019

Journal of Graph Theory

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“…In particular, we will explain in what way our results (Theorems 7 and 9) are essentially best-possible, as well as how we may be able to strengthen these theorems further. Some of our discussion draws on the survey [3].…”

Section: A Discussion On the Optimality Of Degree Sequence Conditionsmentioning

confidence: 99%

“…Definition 34. Let s ∈ N be sufficiently large and λ ∈ R + be sufficiently small where σ(1 + λ)s/ω ∈ N. Recall that σ < ω. LetB be the r-partite bottle graph with neck σ(1 + λ)s/ω and width s. 3 Moreover, we choose λ and s such thatB contains a perfect B-tiling. HenceB contains a perfect H-tiling.…”

Section: Proof Overviewmentioning

confidence: 99%

A Degree Sequence Version of the Kühn–Osthus Tiling Theorem

Hyde

Treglown

2020

Electron. J. Combin.

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“…For instance, minimum degree thresholds for the existence of certain graph structures, such as the threshold for hamiltonicity in Dirac's theorem [12], can be thought of as forcible theorems. Two older but exceptionally thorough surveys on forcible and potential problems are due to Hakimi and Schmeichel [22] and Rao [31], and a more recent survey on forcible "Chvátal-type" theorems (in the spirit of [9]) is due to Bauer et al [3].…”

Section: Introduction a Sequence Of Nonnegative Integersmentioning

confidence: 99%

Extremal Theorems for Degree Sequence Packing and the Two-Color Discrete Tomography Problem

Diemunsch¹,

Ferrara²,

Jahanbekam³

et al. 2015

SIAM J. Discrete Math.

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A sequence π = (d 1 ,. .. , dn) is graphic if there is a simple graph G with vertex set {v 1 ,. .. , vn} such that the degree of v i is d i. We say that graphic sequences π 1 = (d (1) 1 ,. .. , d (1) n) and π 2 = (d (2) 1 ,. .. , d (2) n) pack if there exist edge-disjoint n-vertex graphs G 1 and G 2 such that for j ∈ {1, 2}, d G j (v i) = d (j) i for all i ∈ {1,. .. , n}. Here, we prove several extremal degree sequence packing theorems that parallel central results and open problems from the graph packing literature. Specifically, the main result of this paper implies degree sequence packing analogues to the Bollobás-Eldridge-Catlin graph packing conjecture and the classical graph packing theorem of Sauer and Spencer. In discrete tomography, a branch of discrete imaging science, the goal is to reconstruct discrete objects using data acquired from low-dimensional projections. Specifically, in the k-color discrete tomography problem the goal is to color the entries of an m × n matrix using k colors so that each row and column receives a prescribed number of entries of each color. This problem is equivalent to packing the degree sequences of k bipartite graphs with parts of sizes m and n. Here we also prove several Sauer-Spencer-type theorems with applications to the two-color discrete tomography problem.

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Best Monotone Degree Conditions for Graph Properties: A Survey

Abstract: We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvátal's well-known degree condition for hamiltonicity is best possible.

Cited by 22 publications

References 45 publications

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

A Degree Sequence Version of the Kühn–Osthus Tiling Theorem

Extremal Theorems for Degree Sequence Packing and the Two-Color Discrete Tomography Problem

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