The Degree-Diameter Problem for Sparse Graph Classes

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\lfloor k/2\rfloor})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.

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A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548301004898 ◽

2002 ◽

Vol 11 (1) ◽

pp. 103-111 ◽

Cited By ~ 21

Author(s):

VAN H. VU

Keyword(s):

Chromatic Number ◽

Maximum Degree ◽

Present Author ◽

Upper Bounds ◽

Constant Factor ◽

Chromatic Index ◽

Sparse Graphs ◽

Critical Graphs ◽

Multiplicative Constant ◽

List Chromatic Number

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

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On Ultralimits of Sparse Graph Classes

The Electronic Journal of Combinatorics ◽

10.37236/5519 ◽

2016 ◽

Vol 23 (2) ◽

Author(s):

Michał Pilipczuk ◽

Szymon Toruńczyk

Keyword(s):

Sparse Graph ◽

Sparse Graphs ◽

Dense Graph ◽

Graph Classes

The notion of nowhere denseness is one of the central concepts of the recently developed theory of sparse graphs. We study the properties of nowhere dense graph classes by investigating appropriate limit objects defined using the ultraproduct construction. It appears that different equivalent definitions of nowhere denseness, for example via quasi-wideness or the splitter game, correspond to natural notions for the limit objects that are conceptually simpler and allow for less technically involved reasonings.

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Partitioning Sparse Graphs into an Independent Set and a Forest of Bounded Degree

The Electronic Journal of Combinatorics ◽

10.37236/6815 ◽

2018 ◽

Vol 25 (1) ◽

Author(s):

François Dross ◽

Mickael Montassier ◽

Alexandre Pinlou

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Independent Set ◽

Average Degree ◽

Bounded Degree ◽

Sparse Graphs ◽

Maximum Average Degree

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition. It follows that planar graphs with girth at least $7$ (resp. $8$, $10$) admit an $({\cal I},{\cal F}_5)$-partition (resp. $({\cal I},{\cal F}_3)$-partition, $({\cal I},{\cal F}_2)$-partition).

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A Lower Bound on the Average Degree Forcing a Minor

The Electronic Journal of Combinatorics ◽

10.37236/8847 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Sergey Norin ◽

Bruce Reed ◽

Andrew Thomason ◽

David R. Wood

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Constant Factor ◽

Average Degree ◽

Complete Graphs ◽

Sparse Graphs ◽

Dense Graphs ◽

A Minor

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete graphs by Thomason (2001) and for general dense graphs by Myers and Thomason (2005). It also shows that an upper bound for sparse graphs by Reed and Wood (2016) is best possible up to a constant factor.

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Linear list r-hued coloring of sparse graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500453 ◽

2018 ◽

Vol 10 (04) ◽

pp. 1850045

Author(s):

Hongping Ma ◽

Xiaoxue Hu ◽

Jiangxu Kong ◽

Murong Xu

Keyword(s):

Chromatic Number ◽

Maximum Degree ◽

Average Degree ◽

Disjoint Paths ◽

Proper Coloring ◽

Sparse Graphs ◽

Maximum Average Degree ◽

Linear Formula ◽

Linear List

An [Formula: see text]-hued coloring is a proper coloring such that the number of colors used by the neighbors of [Formula: see text] is at least [Formula: see text]. A linear [Formula: see text]-hued coloring is an [Formula: see text]-hued coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list [Formula: see text]-hued chromatic number, denoted by [Formula: see text], of sparse graphs. It is clear that any graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text]. Let [Formula: see text] be the maximum average degree of a graph [Formula: see text]. In this paper, we obtain the following results: (1) If [Formula: see text], then [Formula: see text] (2) If [Formula: see text], then [Formula: see text]. (3) If [Formula: see text], then [Formula: see text].

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Defective and clustered choosability of sparse graphs

Combinatorics Probability Computing ◽

10.1017/s0963548319000063 ◽

2019 ◽

Vol 28 (5) ◽

pp. 791-810 ◽

Cited By ~ 1

Author(s):

Kevin Hendrey ◽

David R. Wood

Keyword(s):

Graph Theory ◽

Maximum Degree ◽

Discrete Math ◽

Average Degree ◽

Graph Colouring ◽

Sparse Graphs ◽

Trade Off ◽

Maximum Average Degree ◽

The Earth

AbstractAn (improper) graph colouring hasdefect dif each monochromatic subgraph has maximum degree at mostd, and hasclustering cif each monochromatic component has at mostcvertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)kisk-choosable with defectd. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degreem, no (1-ɛ)mbound on the number of colours was previously known. The above result withd=1 solves this problem. It implies that every graph with maximum average degreemis$\lfloor{\frac{3}{4}m+1}\rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu (Discrete Math., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degreemis$\lfloor{\frac{7}{10}m+1}\rfloor$-choosable with clustering 9, and is$\lfloor{\frac{2}{3}m+1}\rfloor$-choosable with clusteringO(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.

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Average Degree Conditions Forcing a Minor

The Electronic Journal of Combinatorics ◽

10.37236/5321 ◽

2016 ◽

Vol 23 (1) ◽

Cited By ~ 4

Author(s):

Daniel J. Harvey ◽

David R. Wood

Keyword(s):

On randomly colouring locally sparse graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.360 ◽

2006 ◽

Vol Vol. 8 ◽

Author(s):

Alan Frieze ◽

Juan Vera

Keyword(s):

Random Graphs ◽

Planar Graphs ◽

Maximum Degree ◽

Glauber Dynamics ◽

Average Degree ◽

Sparse Graphs ◽

International Audience ◽

General Graphs

International audience We consider the problem of generating a random q-colouring of a graph G=(V,E). We consider the simple Glauber Dynamics chain. We show that if for all v ∈ V the average degree of the subgraph H_v induced by the neighbours of v ∈ V is #x226a Δ where Δ is the maximum degree and Δ >c_1\ln n then for sufficiently large c_1, this chain mixes rapidly provided q/Δ >α , where α #x2248 1.763 is the root of α = e^\1/α \. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs G_\n,p\ with p #x226a 1, this beats the 11Δ /6 bound of Vigoda for general graphs.

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