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.

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).

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 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.

scholarly journals Defective and Clustered Graph Colouring

10.37236/7406 ◽  
2018 ◽  
Vol 1000 ◽  
Author(s):  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Graph Colouring ◽  
Outerplanar Graphs ◽  
Open Problems ◽  
Arbitrary Graph ◽  
Maximum Average Degree ◽  
Graph Classes ◽  
Small Defect ◽  
A Minor

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has defect $d$ if each monochromatic component has maximum degree at most $d$. A colouring has clustering $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, graphs with given circumference, graphs excluding a given immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.

scholarly journals New Approach to the $k$-Independence Number of a Graph

10.37236/2646 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Yair Caro ◽  
Adriana Hansberg
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Independence Number ◽  
Independent Set ◽  
Average Degree ◽  
Maximum Cardinality ◽  
New Approach

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of a $k$-independent set of $G$. We prove that, for a graph $G$ on $n$ vertices and average degree $d$, $\alpha_k(G) \ge \frac{k+1}{\lceil d \rceil + k + 1} n$, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on $k$-independence, J. Graph Theory 15 (1991), 99-107].

A maximum degree theorem for diameter-2-critical graphs

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Teresa Haynes ◽  
Michael Henning ◽  
Lucas Merwe ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Discrete Math ◽  
Bipartite Graphs ◽  
Extremal Graphs ◽  
Critical Graphs ◽  
Critical Graph ◽  
Complete Bipartite Graphs

AbstractA graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ⌊n 2/4⌋ and that the extremal graphs are the complete bipartite graphs K ⌊n/2⌋,⌊n/2⌉. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n 0 where n 0 is a tower of 2’s of height about 1014. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.

scholarly journals Separation dimension and degree

2019 ◽  
pp. 1-10
Author(s):  
ALEX SCOTT ◽  
DAVID R. WOOD
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Chromatic Number ◽  
Open Problem ◽  
Maximum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Axis Parallel ◽  
Parallel Hyperplane ◽  
Separation Dimension

Abstract The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝ d , such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].

Injective edge coloring of sparse graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850022
Author(s):  
Yuehua Bu ◽  
Chentao Qi
Keyword(s):  
Upper Bound ◽  
Edge Coloring ◽  
Average Degree ◽  
Sparse Graphs ◽  
Maximum Average Degree ◽  
Coloring Number

A [Formula: see text]-injective edge coloring of a graph [Formula: see text] is a coloring [Formula: see text], such that if [Formula: see text], [Formula: see text] and [Formula: see text] are consecutive edges in [Formula: see text], then [Formula: see text]. [Formula: see text] has a [Formula: see text]-injective edge coloring[Formula: see text] is called the injective edge coloring number. In this paper, we consider the upper bound of [Formula: see text] in terms of the maximum average degree mad[Formula: see text], where [Formula: see text].

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