Decomposing a planar graph without triangular 4-cycles into a matching and a 3-colorable graph

2019 ◽  
Vol 268 ◽  
pp. 112-118 ◽  
Author(s):  
Ziwen Huang ◽  
Runrun Liu ◽  
Gaozhen Wang
Keyword(s):  
Planar Graph ◽  
Colorable Graph

scholarly journals Large independent sets in triangle-free cubic graphs: beyond planarity

2020 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Gwenaël Joret ◽  
Jan Goedgebeur
Keyword(s):  
Planar Graph ◽  
Present Article ◽  
Additional Assumption ◽  
Independence Number ◽  
Independent Set ◽  
Cubic Graphs ◽  
Subcubic Graph ◽  
Color Class ◽  
Subcubic Graphs ◽  
Colorable Graph

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

scholarly journals The Size of Edge-critical Uniquely 3-Colorable Planar Graphs

10.37236/3228 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Naoki Matsumoto
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs ◽  
Infinite Family ◽  
The Other ◽  
Other Hand ◽  
Colorable Graph

A graph $G$ is uniquely $k$-colorable if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. In this paper, we prove that if $G$ is an edge-critical uniquely $3$-colorable planar graph, then $|E(G)|\leq \frac{8}{3}|V(G)|-\frac{17}{3}$. On the other hand, there exists an infinite family of edge-critical uniquely 3-colorable planar graphs with $n$ vertices and $\frac{9}{4}n-6$ edges. Our result gives a first non-trivial upper bound for $|E(G)|$.

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.

scholarly journals A Thomassen-type method for planar graph recoloring

2021 ◽  
Vol 95 ◽  
pp. 103319
Author(s):  
Zdeněk Dvořák ◽  
Carl Feghali
Keyword(s):  
Planar Graph ◽  
Type Method

Plane embeddings of planar graph metrics

2006 ◽  
Author(s):  
MohammadHossein Bateni ◽  
MohammadTaghi Hajiaghayi ◽  
Erik D. Demaine ◽  
Mohammad Moharrami
Keyword(s):  
Planar Graph ◽  
Graph Metrics ◽  
Plane Embeddings

The random division of faces in a planar graph

1996 ◽  
Vol 28 (2) ◽  
pp. 331-331
Author(s):  
Richard Cowan ◽  
Simone Chen
Keyword(s):  
Planar Graph ◽  
Limiting Distribution ◽  
Face Type ◽  
Connected Planar Graph ◽  
Random Division

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

NEIGHBOR SUM DISTINGUISHING COLORING OF SOME GRAPHS

2012 ◽  
Vol 04 (04) ◽  
pp. 1250047 ◽  
Author(s):  
AIJUN DONG ◽  
GUANGHUI WANG
Keyword(s):  
Planar Graph ◽  
Edge Coloring ◽  
Proper Edge Coloring

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.

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