Neighbor Sum Distinguishing Total Colorings of Graphs with Bounded Maximum Degree and 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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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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