Injective edge coloring of sparse graphs

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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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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Strong List Edge Coloring of Subcubic Graphs

Mathematical Problems in Engineering ◽

10.1155/2013/316501 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 3

Author(s):

Hongping Ma ◽

Zhengke Miao ◽

Hong Zhu ◽

Jianhua Zhang ◽

Rong Luo

Keyword(s):

Edge Coloring ◽

Average Degree ◽

Maximum Average Degree ◽

Subcubic Graph ◽

List Edge Coloring ◽

Subcubic Graphs

We study strong list edge coloring of subcubic graphs, and we prove that every subcubic graph with maximum average degree less than 15/7, 27/11, 13/5, and 36/13 can be strongly list edge colored with six, seven, eight, and nine colors, respectively.

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