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

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.

scholarly journals A short note on conflict‐free coloring on closed neighborhoods of bounded degree graphs

2021 ◽  
Author(s):  
Sriram Bhyravarapu ◽  
Subrahmanyam Kalyanasundaram ◽  
Rogers Mathew
Keyword(s):  
Short Note ◽  
Bounded Degree ◽  
Bounded Degree Graphs

scholarly journals EMBEDDING SPANNING BOUNDED DEGREE GRAPHS IN RANDOMLY PERTURBED GRAPHS

Mathematika ◽  
2020 ◽  
Vol 66 (2) ◽  
pp. 422-447 ◽  
Author(s):  
Julia Böttcher ◽  
Richard Montgomery ◽  
Olaf Parczyk ◽  
Yury Person
Keyword(s):  
Bounded Degree ◽  
Bounded Degree Graphs

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.

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