scholarly journals Planar Graphs with the Distance of 6--Cycles at Least 2 from Each Other Are DP-3-Colorable

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 70
Author(s):  
Yueying Zhao ◽  
Lianying Miao
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
List Coloring

DP-coloring as a generalization of list coloring was introduced by Dvořák and Postle recently. In this paper, we prove that every planar graph in which the distance between 6−-cycles is at least 2 is DP-3-colorable, which extends the result of Yin and Yu [Discret. Math. 2019, 342, 2333–2341].

scholarly journals Distributed Coloring in Sparse Graphs with Fewer Colors

10.37236/8395 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Pierre Aboulker ◽  
Marthe Bonamy ◽  
Nicolas Bousquet ◽  
Louis Esperet
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Distributed Algorithm ◽  
Planar Graphs ◽  
List Coloring ◽  
Sparse Graphs ◽  
Four Color Theorem

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in $\text{polylog}(n)$ rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.

scholarly journals Path Choosability of Planar Graphs

10.37236/7139 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Glenn G. Chappell ◽  
Chris Hartman
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Vertex Coloring ◽  
List Coloring ◽  
Outerplanar Graphs ◽  
Color Class ◽  
Path Coloring

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

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].

(1,0)-Relaxed strong edge list coloring of planar graphs with girth 6

2019 ◽  
Vol 11 (06) ◽  
pp. 1950064
Author(s):  
Kai Lin ◽  
Min Chen ◽  
Dong Chen
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
List Coloring ◽  
Chromatic Index ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
Number Formula ◽  
Edge List

Let [Formula: see text] be a graph. An [Formula: see text]-relaxed strong edge [Formula: see text]-coloring is a mapping [Formula: see text] such that for any edge [Formula: see text], there are at most [Formula: see text] edges adjacent to [Formula: see text] and [Formula: see text] edges which are distance two apart from [Formula: see text] assigned the same color as [Formula: see text]. The [Formula: see text]-relaxed strong chromatic index, denoted by [Formula: see text], is the minimum number [Formula: see text] of an [Formula: see text]-relaxed strong [Formula: see text]-edge-coloring admitted by [Formula: see text]. [Formula: see text] is called [Formula: see text]-relaxed strong edge [Formula: see text]-colorable if for a given list assignment [Formula: see text], there exists an [Formula: see text]-relaxed strong edge coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-colorable for any list assignment with [Formula: see text] for all [Formula: see text], then [Formula: see text] is said to be [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. The [Formula: see text]-relaxed strong list chromatic index, denoted by [Formula: see text], is defined to be the smallest integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. In this paper, we prove that every planar graph [Formula: see text] with girth 6 satisfies that [Formula: see text]. This strengthens a result which says that every planar graph [Formula: see text] with girth 7 and [Formula: see text] satisfies that [Formula: see text].

scholarly journals 3-list-coloring planar graphs of girth 4

2011 ◽  
Vol 311 (6) ◽  
pp. 413-417 ◽  
Author(s):  
Jun-Lin Guo ◽  
Yue-Li Wang
Keyword(s):  
Planar Graphs ◽  
List Coloring

scholarly journals KNOT-INEVITABLE PROJECTIONS OF PLANAR GRAPHS

1996 ◽  
Vol 05 (06) ◽  
pp. 877-883 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
TATSUYA TSUKAMOTO
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Spatial Graph ◽  
Torus Knot ◽  
Nonplanar Graph

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

scholarly journals Planar Graphs of Maximum Degree Six without 7-cycles are Class One

10.37236/2589 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Danjun Huang ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

In this paper, we prove that every planar graph of maximum degree six without 7-cycles is class one.

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

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