Spanning Trees in Dense Graphs

In this paper we prove the following almost optimal theorem. For any δ > 0, there exist constants c and n0 such that, if n [ges ] n0, T is a tree of order n and maximum degree at most cn/log n, and G is a graph of order n and minimum degree at least (1/2 + δ)n, then T is a subgraph of G.

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The bandwidth theorem for locally dense graphs

Forum of Mathematics Sigma ◽

10.1017/fms.2020.39 ◽

2020 ◽

Vol 8 ◽

Author(s):

Katherine Staden ◽

Andrew Treglown

Keyword(s):

Maximum Degree ◽

Blow Up ◽

Minimum Degree ◽

Bounded Degree ◽

Dense Graphs ◽

Vertex Graph ◽

Delta G

Abstract The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás andKomlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. In this paper, we prove a version of the bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense n-vertex graph G with $\delta (G)> (1/2+o(1))n$ contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.

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Every 3-polytope with minimum degree 5 has a 6-cycle with maximum degree at most 11

Discrete Mathematics ◽

10.1016/j.disc.2013.10.021 ◽

2014 ◽

Vol 315-316 ◽

pp. 128-134 ◽

Cited By ~ 4

Author(s):

O.V. Borodin ◽

A.O. Ivanova ◽

A.V. Kostochka

Keyword(s):

Maximum Degree ◽

Minimum Degree

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Semi-Deterministic Construction of Scale-Free Networks with Designated Parameters

Journal of Interconnection Networks ◽

10.1142/s0219265918500019 ◽

2018 ◽

Vol 18 (01) ◽

pp. 1850001

Author(s):

NAOKI TAKEUCHI ◽

SATOSHI FUJITA

Keyword(s):

Degree Distribution ◽

Interconnection Networks ◽

Maximum Degree ◽

Minimum Degree ◽

Scale Free ◽

Scale Free Networks ◽

Ideal Number ◽

Message Propagation ◽

Deterministic Construction ◽

The Ideal

Scale-free networks have several favorable properties as the topology of interconnection networks such as the short diameter and the quick message propagation. In this paper, we propose a method to construct scale-free networks in a semi-deterministic manner. The proposed algorithm extends the Bulut's algorithm for constructing scale-free networks with designated minimum degree k and maximum degree m, in such a way that: (1) it determines the ideal number of edges derived from the ideal degree distribution; and (2) after connecting each new node to k existing nodes as in the Bulut’s algorithm, it adjusts the number of edges to the ideal value by conducting add/removal of edges. We prove that such an adjustment is always possible if the number of nodes in the network exceeds [Formula: see text]. The performance of the algorithm is experimentally evaluated.

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On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

The Electronic Journal of Combinatorics ◽

10.37236/5173 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 8

Author(s):

Jakub Przybyło

Keyword(s):

Connected Graph ◽

Special Class ◽

Maximum Degree ◽

Minimum Degree ◽

Probabilistic Approach ◽

The Family ◽

Edge Colourings ◽

Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

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Pitchfork domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500251 ◽

2020 ◽

Vol 12 (02) ◽

pp. 2050025

Author(s):

Manal N. Al-Harere ◽

Mohammed A. Abdlhusein

Keyword(s):

Maximum Degree ◽

Undirected Graph ◽

Minimum Degree ◽

Dominating Set ◽

Domination Number ◽

Dominating Sets ◽

Minimum Cardinality ◽

New Model ◽

Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

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Spanning k-tree with specified vertices

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500435 ◽

2019 ◽

Vol 11 (04) ◽

pp. 1950043

Author(s):

Feifei Song ◽

Jianjie Zhou

Keyword(s):

Maximum Degree ◽

Spanning Trees ◽

Independent Set ◽

Disjoint Paths ◽

Maximum Independent Set ◽

Local Connectivity

A [Formula: see text]-tree is a tree with maximum degree at most [Formula: see text]. For a graph [Formula: see text] and [Formula: see text] with [Formula: see text], let [Formula: see text] be the cardinality of a maximum independent set containing [Formula: see text] and [Formula: see text]. For a graph [Formula: see text] and [Formula: see text], the local connectivity [Formula: see text] is defined to be the maximum number of internally disjoint paths connecting [Formula: see text] and [Formula: see text] in [Formula: see text]. In this paper, we prove the following theorem and show the condition is sharp. Let [Formula: see text], [Formula: see text] and [Formula: see text] be integers with [Formula: see text], [Formula: see text] and [Formula: see text]. For any two nonadjacent vertices [Formula: see text] and [Formula: see text] of [Formula: see text], we have [Formula: see text] and [Formula: see text]. Then for any [Formula: see text] distinct vertices of [Formula: see text], [Formula: see text] has a spanning [Formula: see text]-tree such that each of [Formula: see text] specified vertices has degree at most [Formula: see text]. This theorem implies H. Matsuda and H. Matsumura’s result in [on a [Formula: see text]-tree containing specified cleares in a graph, Graphs Combin. 22 (2006) 371–381] and V. Neumann-Lara and E. Rivera-Campo’s result in [Spanning trees with bounded degrees, Combinatorica 11 (1991) 55–61].

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On the maximum degree of minimum spanning trees

Proceedings of the tenth annual symposium on Computational geometry - SCG '94 ◽

10.1145/177424.177978 ◽

1994 ◽

Cited By ~ 9

Author(s):

Gabriel Robins ◽

Jeffrey S. Salowe

Keyword(s):

Maximum Degree ◽

Spanning Trees ◽

Minimum Spanning Trees

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Tilings in Randomly Perturbed Dense Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548318000366 ◽

2018 ◽

Vol 28 (2) ◽

pp. 159-176 ◽

Cited By ~ 12

Author(s):

JÓZSEF BALOGH ◽

ANDREW TREGLOWN ◽

ADAM ZSOLT WAGNER

Keyword(s):

High Probability ◽

Minimum Degree ◽

Graph Model ◽

Regularity Lemma ◽

Arbitrary Graph ◽

Random Graph Model ◽

Dense Graphs ◽

Szemeredi's Regularity Lemma ◽

Vertex Disjoint ◽

Special Case

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

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