scholarly journals Bounds on the Geometric Mean of Arc Lengths for Bounded-Degree Planar Graphs

2009 ◽  
pp. 153-162
Author(s):  
Mohammad Khairul Hasan ◽  
Sung-Eui Yoon ◽  
Kyung-Yong Chwa
Keyword(s):  
Planar Graphs ◽  
Geometric Mean ◽  
Bounded Degree ◽  
Arc Lengths

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 $\Delta$ .

Planar Graphs of Bounded Degree Have Bounded Queue Number

2019 ◽  
Vol 48 (5) ◽  
pp. 1487-1502 ◽  
Author(s):  
Michael A. Bekos ◽  
Henry Förster ◽  
Martin Gronemann ◽  
Tamara Mchedlidze ◽  
Fabrizio Montecchiani ◽  
...  
Keyword(s):  
Planar Graphs ◽  
Bounded Degree

scholarly journals Corrigendum to “Complexity analysis of P3-convexity problems on bounded-degree and planar graphs” [Theoret. Comput. Sci. 607 Part 1 (2015) 83–95]

2017 ◽  
Vol 704 ◽  
pp. 92-93
Author(s):  
Lucia Draque Penso ◽  
Fábio Protti ◽  
Dieter Rautenbach ◽  
Uéverton dos Santos Souza
Keyword(s):  
Planar Graphs ◽  
Complexity Analysis ◽  
Bounded Degree

scholarly journals Hypocomb: Bounded-Degree Localized Geometric Planar Graphs for Wireless Ad Hoc Networks

2013 ◽  
Vol 24 (7) ◽  
pp. 1341-1354 ◽  
Author(s):  
Xu Li ◽  
Nathalie Mitton ◽  
Isabelle Simplot-Ryl ◽  
David Simplot-Ryl
Keyword(s):  
Ad Hoc Networks ◽  
Wireless Ad Hoc Networks ◽  
Planar Graphs ◽  
Ad Hoc ◽  
Bounded Degree ◽  
Hoc Networks

scholarly journals Universal Graphs for Bounded-Degree Trees and Planar Graphs

1989 ◽  
Vol 2 (2) ◽  
pp. 145-155 ◽  
Author(s):  
Sandeep N. Bhatt ◽  
F. R. K. Chung ◽  
F. T. Leighton ◽  
Arnold L. Rosenberg
Keyword(s):  
Planar Graphs ◽  
Bounded Degree ◽  
Universal Graphs ◽  
Bounded Degree Trees

scholarly journals Adjacency Labelling for Planar Graphs (and Beyond)

2021 ◽  
Vol 68 (6) ◽  
pp. 1-33
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Cyril Gavoille ◽  
Gwenaël Joret ◽  
Piotr Micek ◽  
...  
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Order Term ◽  
Optimal Result ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Graph Classes ◽  
Bounded Degree Graphs ◽  
Minor Free Graphs ◽  
Labelling Scheme

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n -vertex planar graph G is assigned a (1 + o(1)) log 2 n -bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G . This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n , there exists a graph U n with n 1+o(1) vertices such that every n -vertex planar graph is an induced subgraph of U n . These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k -planar graphs.

scholarly journals Partitioning planar graphs without 4-cycles and 5-cycles into bounded degree forests

2021 ◽  
Vol 344 (1) ◽  
pp. 112172
Author(s):  
Eun-Kyung Cho ◽  
Ilkyoo Choi ◽  
Boram Park
Keyword(s):  
Planar Graphs ◽  
Bounded Degree

Bounded-Degree Independent Sets in Planar Graphs

2005 ◽  
Vol 38 (3) ◽  
pp. 253-278 ◽  
Author(s):  
Therese Biedl ◽  
Dana F. Wilkinson
Keyword(s):  
Planar Graphs ◽  
Independent Sets ◽  
Bounded Degree
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