Neighbor Sum Distinguishing Total Choice Number of Planar Graphs without 6-cycles

2020 ◽  
Vol 36 (12) ◽  
pp. 1417-1428
Author(s):  
Dong Han Zhang ◽  
You Lu ◽  
Sheng Gui Zhang
Keyword(s):  
Planar Graphs ◽  
Choice Number

Two results on K-(2,1)-total choosability of planar graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050081
Author(s):  
Yan Song ◽  
Lei Sun
Keyword(s):  
Planar Graph ◽  
Euler Characteristic ◽  
Planar Graphs ◽  
Choice Number

The [Formula: see text]-total choice number of [Formula: see text], denoted by [Formula: see text], is the minimum [Formula: see text] such that [Formula: see text] is [Formula: see text]-[Formula: see text]-total choosable. It was proved in [Y. Yu, X. Zhang and G. Z. Liu, List (d,1)-total labeling of graphs embedded in surfaces, Oper. Res. Trans. 15(3) (2011) 29–37.] that [Formula: see text] if [Formula: see text] is a graph embedded in surface with Euler characteristic [Formula: see text] and [Formula: see text] big enough. In this paper, we prove that: (i) if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text]-cycle is not adjacent to [Formula: see text]-cycle, [Formula: see text], then [Formula: see text]; (ii) if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text]-cycle is not adjacent to [Formula: see text]-cycle, where [Formula: see text], then [Formula: see text].

2-distance choice number of planar graphs with maximal degree no more than 4

2020 ◽  
pp. 2150051
Author(s):  
Wenjuan Zhou ◽  
Lei Sun
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Maximal Degree ◽  
Choice Number

Regarding the 2-[Formula: see text] coloring of planar graphs, in 1977, Wegner conjectured that for a graph [Formula: see text]: (1) [Formula: see text] if [Formula: see text]. (2) [Formula: see text] if [Formula: see text]. (3) [Formula: see text] if [Formula: see text]. In this paper, we proved that for every planar graph with maximum degree [Formula: see text]: (1) [Formula: see text] if [Formula: see text]. (2) [Formula: see text] if [Formula: see text].

Planar graphs without intersecting 5-cycles are signed-4-choosable

2021 ◽  
pp. 2150151
Author(s):  
Seog-Jin Kim ◽  
Xiaowei Yu
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Discrete Math ◽  
Signed Graph ◽  
List Coloring ◽  
Minimum Integer ◽  
Number Formula ◽  
Choice Number ◽  
The Difference ◽  
Signature Formula

A signed graph is a pair [Formula: see text], where [Formula: see text] is a graph and [Formula: see text] is a signature of [Formula: see text]. A set [Formula: see text] of integers is symmetric if [Formula: see text] implies that [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an [Formula: see text]-coloring of a signed graph [Formula: see text] is a coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text] and [Formula: see text] for every edge [Formula: see text]. The signed choice number [Formula: see text] of a graph [Formula: see text] is defined to be the minimum integer [Formula: see text] such that for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] and for any signature [Formula: see text] on [Formula: see text], there is a proper [Formula: see text]-coloring of [Formula: see text]. List signed coloring is a generalization of list coloring. However, the difference between signed choice number and choice number can be arbitrarily large. Hu and Wu [Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792] showed that every planar graph without intersecting 5-cycles is 4-choosable. In this paper, we prove that [Formula: see text] if [Formula: see text] is a planar graph without intersecting 5-cycles, which extends the main result of [D. Hu and J. Wu, Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792].

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

A Space-Efficient Separator Algorithm for Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 1007-1016 ◽  
Author(s):  
Ryo ASHIDA ◽  
Sebastian KUHNERT ◽  
Osamu WATANABE
Keyword(s):  
Planar Graphs

Note on (semi-)proper orientation of some triangulated planar graphs

2021 ◽  
Vol 392 ◽  
pp. 125723
Author(s):  
Ruijuan Gu ◽  
Hui Lei ◽  
Yulai Ma ◽  
Zhenyu Taoqiu
Keyword(s):  
Planar Graphs ◽  
Proper Orientation

scholarly journals Distributed Dominating Set Approximations beyond Planar Graphs

2019 ◽  
Vol 15 (3) ◽  
pp. 1-18 ◽  
Author(s):  
Saeed Akhoondian Amiri ◽  
Stefan Schmid ◽  
Sebastian Siebertz
Keyword(s):  
Planar Graphs ◽  
Dominating Set

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

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