Contributions to the theory of graphic sequences

Zverovich, Igor E.; Zverovich, Vadim

doi:10.1016/0012-365x(92)90152-6

Cited by 39 publications

(38 citation statements)

References 6 publications

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“…(4) gives in k m , is attained at k m = 1 2 (a + b + 1 ± 1) and is equal to k(n − 1) + 1 4 ((a + b + 1) 2 − 1) − bn. Thus the inequality (a + b + 1) 2 ≤ 4bn + 1 implies r k ≤ k(n − 1) and so the sequence d is graphic by [8,Theorem 3]. In case (II), the (unique) maximal value of the right-hand side of Eq.…”

Section: Lemma 6 For a Strong Index Kmentioning

confidence: 94%

“…Sufficiency in case (I) follows immediately from Theorem 1. For the other cases, we employ similar ideas to those of [8].…”

Section: Proof Of Sufficiencymentioning

confidence: 96%

“…(3) becomes strict, hence r k ≤ k(n − 1) + 1 4 (a + b + 1) 2 − bn − 1. Thus the inequality (a + b + 1) 2 ≤ 4bn + 4 implies r k ≤ k(n − 1) and therefore the sequence d is graphic by [8,Theorem 3].…”

Section: Lemma 6 For a Strong Index Kmentioning

confidence: 97%

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A sharp refinement of a result of Zverovich–Zverovich

Cairns

Mendan

Nikolayevsky

2015

Discrete Mathematics

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a b s t r a c tFor a finite sequence of positive integers to be the degree sequence of a finite graph, Zverovich and Zverovich gave a sufficient condition involving only the length of the sequence, its largest entry and its smallest entry. Barrus, Hartke, Jao, and West gave a better bound and showed that the sharp bound is never less than theirs by more than 1. In this paper we determine the sharp bound.

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Section: Lemma 6 For a Strong Index Kmentioning

confidence: 94%

“…Sufficiency in case (I) follows immediately from Theorem 1. For the other cases, we employ similar ideas to those of [8].…”

Section: Proof Of Sufficiencymentioning

confidence: 96%

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A sharp refinement of a result of Zverovich–Zverovich

Cairns

Mendan

Nikolayevsky

2015

Discrete Mathematics

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“…Proof. The first term in this bound comes by substituting α 1 + 1 for the maximum degree into the graphic bound given by Zverovich and Zverovich (Theorem 6, [10]); it follows that if the above bound holds then ⊕ 1,2 α is graphic. The second term uses a degree sequence packing result.…”

Section: Forced and Forbidden Edgesmentioning

confidence: 99%

Forced Edges and Graph Structure

Cloteaux

2019

J. RES. NATL. INST. STAN.

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For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either forced or forbidden edges. Among the things we show, we determine the structure of the forced or forbidden edge sets, the relationship between the sizes of forced and forbidden sets for a sequence, and the resulting structural consequences to their realizations. This includes showing that the diameter of every realization of a degree sequence containing forced or forbidden edges is no greater than 3, and that these graphs are maximally edge-connected. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

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“…In Section 3, we give sufficient conditions based on the maximum degree in E 12 for a solution to exist for the case of non oriented complete bipartite graphs and complete graphs (see [18] for additional results on the sequences of degrees of a graph). This exhibits a solvable case of the basic image reconstruction problem with k = 3 colors.…”

Section: Introductionmentioning

confidence: 99%

Degree-constrained edge partitioning in graphs arising from discrete tomography

Bentz¹,

Costa²,

Picouleau³

et al. 2009

JGAA

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Starting from the basic problem of reconstructing a 2-dimensional image given by its projections on two axes, one associates a model of edge coloring in a complete bipartite graph. The complexity of the case with k = 3 colors is open. Variations and special cases are considered for the case k = 3 colors where the graph corresponding to the union of some color classes (for instance colors 1 and 2) has a given structure (tree, vertexdisjoint chains, 2-factor, etc.). We also study special cases corresponding to the search of 2 edge-disjoint chains or cycles going through specified vertices. A variation where the graph is oriented is also presented.In addition we explore similar problems for the case where the underlying graph is a complete graph (instead of a complete bipartite graph).

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Contributions to the theory of graphic sequences

Cited by 39 publications

References 6 publications

A sharp refinement of a result of Zverovich–Zverovich

A sharp refinement of a result of Zverovich–Zverovich

Forced Edges and Graph Structure

Degree-constrained edge partitioning in graphs arising from discrete tomography

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