Decomposition of sparse graphs into two forests, one having bounded maximum degree

2010 ◽  
Vol 110 (20) ◽  
pp. 913-916 ◽  
Author(s):  
Mickael Montassier ◽  
André Raspaud ◽  
Xuding Zhu
Keyword(s):  
Maximum Degree ◽  
Sparse Graphs

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

2002 ◽  
Vol 11 (1) ◽  
pp. 103-111 ◽  
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.

scholarly journals The Degree-Diameter Problem for Sparse Graph Classes

10.37236/4313 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Guillermo Pineda-Villavicencio ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Sparse Graphs ◽  
Graph Classes ◽  
Minor Free Graphs

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.

On Brooks' Theorem for Sparse Graphs

1995 ◽  
Vol 4 (2) ◽  
pp. 97-132 ◽  
Author(s):  
Jeong Han Kim
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Sparse Graphs ◽  
Choice Number ◽  
Image Position

Let G be a graph with maximum degree Δ(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfieswhere o(l) goes to zero as Δ(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number:provided g(G) < 4.

scholarly journals Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

2009 ◽  
Vol 65 (2) ◽  
pp. 83-93 ◽  
Author(s):  
O. V. Borodin ◽  
A. O. Ivanova ◽  
M. Montassier ◽  
P. Ochem ◽  
A. Raspaud
Keyword(s):  
Maximum Degree ◽  
Sparse Graphs

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

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).

scholarly journals Intersections of Randomly Embedded Sparse Graphs are Poisson

10.37236/1468 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Edward A. Bender ◽  
E. Rodney Canfield
Keyword(s):  
Maximum Degree ◽  
Spanning Trees ◽  
Sufficient Conditions ◽  
Intersection Theorem ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Inversion Theorem ◽  
Number Of Spanning Trees

Suppose that $t \ge 2$ is an integer, and randomly label $t$ graphs with the integers $1 \dots n$. We give sufficient conditions for the number of edges common to all $t$ of the labelings to be asymptotically Poisson as $n \to \infty$. We show by example that our theorem is, in a sense, best possible. For $G_n$ a sequence of graphs of bounded degree, each having at most $n$ vertices, Tomescu has shown that the number of spanning trees of $K_n$ having $k$ edges in common with $G_n$ is asymptotically $e^{-2s/n}(2s/n)^k/k! \times n^{n-2}$, where $s=s(n)$ is the number of edges in $G_n$. As an application of our Poisson-intersection theorem, we extend this result to the case in which maximum degree is only restricted to be ${\scriptstyle\cal O}(n \log\log n/\log n)$. We give an inversion theorem for falling moments, which we use to prove our Poisson-intersection theorem.

Total-coloring of Sparse Graphs with Maximum Degree 6

2021 ◽  
Vol 37 (4) ◽  
pp. 738-746
Author(s):  
Yu-lin Chang ◽  
Fei Jing ◽  
Guang-hui Wang ◽  
Ji-chang Wu
Keyword(s):  
Maximum Degree ◽  
Total Coloring ◽  
Sparse Graphs

Linear list r-hued coloring of sparse graphs

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].

scholarly journals Defective and clustered choosability of sparse graphs

2019 ◽  
Vol 28 (5) ◽  
pp. 791-810 ◽  
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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