Small Universal Graphs for Bounded-Degree Planar Graphs

COMBINATORICA ◽  
2002 ◽  
Vol 22 (3) ◽  
pp. 345-359 ◽  
Author(s):  
Michael Capalbo
Keyword(s):  
Planar Graphs ◽  
Bounded Degree ◽  
Universal Graphs

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 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 Optimal induced universal graphs for bounded-degree graphs

2017 ◽  
Vol 166 (1) ◽  
pp. 61-74 ◽  
Author(s):  
NOGA ALON ◽  
RAJKO NENADOV
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Vertex Graph ◽  
Bounded Degree Graphs ◽  
Universal Graphs

AbstractWe show that for any constant Δ ≥ 2, there exists a graph Γ withO(nΔ / 2) vertices which contains everyn-vertex graph with maximum degree Δ as an induced subgraph. For odd Δ this significantly improves the best-known earlier bound and is optimal up to a constant factor, as it is known that any such graph must have at least Ω(nΔ/2) vertices.

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 On induced-universal graphs for the class of bounded-degree graphs

2008 ◽  
Vol 108 (5) ◽  
pp. 255-260 ◽  
Author(s):  
Louis Esperet ◽  
Arnaud Labourel ◽  
Pascal Ochem
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Graphs ◽  
Universal Graphs

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
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