scholarly journals The Glauber Dynamics for Colourings of Bounded Degree Trees

2009 ◽  
pp. 631-645 ◽  
Author(s):  
Brendan Lucier ◽  
Michael Molloy ◽  
Yuval Peres
Keyword(s):  
Glauber Dynamics ◽  
Bounded Degree ◽  
Bounded Degree Trees

The Glauber Dynamics for Colorings of Bounded Degree Trees

2011 ◽  
Vol 25 (2) ◽  
pp. 827-853 ◽  
Author(s):  
B. Lucier ◽  
M. Molloy
Keyword(s):  
Glauber Dynamics ◽  
Bounded Degree ◽  
Bounded Degree Trees

Tight Bounds for Embedding Bounded Degree Trees

2010 ◽  
pp. 95-137 ◽  
Author(s):  
Béla Csaba ◽  
Judit Nagy-György ◽  
Ian Levitt ◽  
Endre Szemerédi
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Trees

Very rapid mixing of the Glauber dynamics for proper colorings on bounded-degree graphs

2002 ◽  
Vol 20 (1) ◽  
pp. 98-114 ◽  
Author(s):  
Martin Dyer ◽  
Catherine Greenhill ◽  
Mike Molloy
Keyword(s):  
Glauber Dynamics ◽  
Bounded Degree ◽  
Rapid Mixing ◽  
Bounded Degree Graphs

scholarly journals Helly EPT graphs on bounded degree trees: Characterization and recognition

2017 ◽  
Vol 340 (12) ◽  
pp. 2798-2806
Author(s):  
L. Alcón ◽  
M. Gutierrez ◽  
M.P. Mazzoleni
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Trees

scholarly journals Ramsey Goodness of Bounded Degree Trees

2018 ◽  
Vol 27 (3) ◽  
pp. 289-309 ◽  
Author(s):  
IGOR BALLA ◽  
ALEXEY POKROVSKIY ◽  
BENNY SUDAKOV
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Colour Class ◽  
Bounded Degree Trees

Given a pair of graphsGandH, the Ramsey numberR(G,H) is the smallestNsuch that every red–blue colouring of the edges of the complete graphKNcontains a red copy ofGor a blue copy ofH. If a graphGis connected, it is well known and easy to show thatR(G,H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number ofHand σ(H) is the size of the smallest colour class in a χ(H)-colouring ofH. A graphGis calledH-goodifR(G,H) = (|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then.In this paper we show that ifn≥ Ω(|H| log4|H|) then everyn-vertex bounded degree treeTisH-good. The dependency betweennand |H| is tight up to log factors. This substantially improves a result of Erdős, Faudree, Rousseau, and Schelp from 1985, who proved thatn-vertex bounded degree trees areH-good whenn≥ Ω(|H|4).

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 Optimal packings of bounded degree trees

2019 ◽  
Vol 21 (12) ◽  
pp. 3573-3647 ◽  
Author(s):  
Felix Joos ◽  
Jaehoon Kim ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Trees

Frozen (Δ + 1)-colourings of bounded degree graphs

2020 ◽  
pp. 1-14
Author(s):  
Marthe Bonamy ◽  
Nicolas Bousquet ◽  
Guillem Perarnau
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Mixing Time ◽  
Glauber Dynamics ◽  
Regular Graphs ◽  
Connected Component ◽  
Bounded Degree ◽  
Large Girth ◽  
Bounded Degree Graphs

Abstract Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices. In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

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