Universal Graphs for Bounded-Degree Trees and Planar Graphs

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 $\Delta$ .

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Tight Bounds for Embedding Bounded Degree Trees

Fete of Combinatorics and Computer Science - Bolyai Society Mathematical Studies ◽

10.1007/978-3-642-13580-4_5 ◽

2010 ◽

pp. 95-137 ◽

Cited By ~ 6

Author(s):

Béla Csaba ◽

Judit Nagy-György ◽

Ian Levitt ◽

Endre Szemerédi

Keyword(s):

Bounded Degree ◽

Bounded Degree Trees

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The Complexity of the Cover Polynomials for Planar Graphs of Bounded Degree

Mathematical Foundations of Computer Science 2011 - Lecture Notes in Computer Science ◽

10.1007/978-3-642-22993-0_12 ◽

2011 ◽

pp. 96-107 ◽

Cited By ~ 1

Author(s):

Markus Bläser ◽

Radu Curticapean

Keyword(s):

Planar Graphs ◽

Bounded Degree

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Planar Graphs of Bounded Degree Have Bounded Queue Number

SIAM Journal on Computing ◽

10.1137/19m125340x ◽

2019 ◽

Vol 48 (5) ◽

pp. 1487-1502 ◽

Cited By ~ 3

Author(s):

Michael A. Bekos ◽

Henry Förster ◽

Martin Gronemann ◽

Tamara Mchedlidze ◽

Fabrizio Montecchiani ◽

...

Keyword(s):

Planar Graphs ◽

Bounded Degree

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Optimal induced universal graphs for bounded-degree graphs

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000706 ◽

2017 ◽

Vol 166 (1) ◽

pp. 61-74 ◽

Cited By ~ 1

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.

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