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 A Linear-Time Algorithm for 4-Coloring Some Classes of Planar Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Zuosong Liang ◽  
Huandi Wei
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Vertex Coloring ◽  
Computer Assisted ◽  
Linear Time Algorithm ◽  
Four Color Theorem ◽  
Finite Set ◽  
Proper Vertex

Every graph G = V , E considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e ∈ E of G with an unordered pair of vertices of G which are called the ends of the edge e . A graph is said to be a planar graph if it can be drawn in the plane so that its edges intersect only at their ends. A proper k -vertex-coloring of a graph G = V , E is a mapping c : V ⟶ S ( S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar graph has a proper vertex-coloring with four colors. However, the current known proof for the Four Color Theorem is computer assisted. In addition, the correctness of the proof is still lengthy and complicated. In 2010, a simple O n 2 time algorithm was provided to 4-color a 3-colorable planar graph. In this paper, we give an improved linear-time algorithm to either output a proper 4-coloring of G or conclude that G is not 3-colorable when an arbitrary planar graph G is given. Using this algorithm, we can get the proper 4-colorings of 3-colorable planar graphs, planar graphs with maximum degree at most five, and claw-free planar graphs.

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 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.

scholarly journals On $t$-Common List-Colorings

10.37236/6738 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Hojin Choi ◽  
Young Soo Kwon
Keyword(s):  
Positive Integer ◽  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Planar Graphs ◽  
Four Color Theorem ◽  
List Colorings ◽  
List Chromatic Number ◽  
Positive Integers

In this paper, we introduce a new variation of list-colorings. For a graph $G$  and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) =  \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

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].

scholarly journals On the Strong Chromatic Index of Sparse Graphs

10.37236/5390 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Philip DeOrsey ◽  
Michael Ferrara ◽  
Nathan Graber ◽  
Stephen G. Hartke ◽  
Luke L. Nelsen ◽  
...  
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Edge Coloring ◽  
Combinatorial Nullstellensatz ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Strong Chromatic Index ◽  
General Bound

The strong chromatic index of a graph $G$, denoted $\chi'_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi'_{s,\ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with ${\rm girth}(G) \geq 41$ then $\chi'_{s,\ell}(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and ${\rm girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper.Finally, if $G$ is a planar graph with maximum degree at most four and ${\rm girth}(G) \geq 28$, then $\chi'_s(G)N \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its applications to the strong edge coloring, Applied Mathematics and Computation, 325 (2018), 246-251] in this case.

scholarly journals Colorings and Orientations of Matrices and Graphs

10.37236/1087 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Planar Graphs ◽  
Partial Solution ◽  
General Concept ◽  
List Coloring ◽  
Weighted Sum ◽  
Incidence Matrices ◽  
Graph Theoretic ◽  
Four Color Theorem ◽  
List Colorings

We introduce colorings and orientations of matrices as generalizations of the graph theoretic terms. The permanent per$(A[\zeta|\xi])$ of certain copies $A[\zeta|\xi]$ of a matrix $A$ can be expressed as a weighted sum over the orientations or the colorings of $A$. When applied to incidence matrices of graphs these equations include Alon and Tarsi's theorem about Eulerian orientations and the existence of list colorings. In the case of planar graphs we deduce Ellingham and Goddyn's partial solution of the list coloring conjecture and Scheim's equivalency between not vanishing permanents and the four color theorem. The general concept of matrix colorings in the background is also connected to hypergraph colorings and matrix choosability.

(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 A Note on Not-4-List Colorable Planar Graphs

10.37236/7320 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Margit Voigt ◽  
Arnfried Kemnitz
Keyword(s):  
Planar Graph ◽  
Bipartite Graph ◽  
Planar Graphs ◽  
The Other ◽  
Four Color Theorem ◽  
List Colorings ◽  
Vertex Set ◽  
Colorable Graph ◽  
Negative Strengthening

The Four Color Theorem states that every planar graph is properly 4-colorable. Moreover, it is well known that there are planar graphs that are non-$4$-list colorable. In this paper we investigate a problem combining proper colorings and list colorings. We ask whether the vertex set of every planar graph can be partitioned into two subsets where one subset induces a bipartite graph and the other subset induces a $2$-list colorable graph. We answer this question in the negative strengthening the result on non-$4$-list colorable planar graphs.

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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