Locally Dense Independent Sets in Regular Graphs of Large Girth—An Example of a New Approach

2008 ◽  
pp. 163-183 ◽  
Author(s):  
Frank Göring ◽  
Jochen Harant ◽  
Dieter Rautenbach ◽  
Ingo Schiermeyer
Keyword(s):  
Independent Sets ◽  
Regular Graphs ◽  
New Approach ◽  
Large Girth

scholarly journals Invariant Gaussian processes and independent sets on regular graphs of large girth

2014 ◽  
Vol 47 (2) ◽  
pp. 284-303 ◽  
Author(s):  
Endre Csóka ◽  
Balázs Gerencsér ◽  
Viktor Harangi ◽  
Bálint Virág
Keyword(s):  
Gaussian Processes ◽  
Independent Sets ◽  
Regular Graphs ◽  
Large Girth

scholarly journals Large independent sets in regular graphs of large girth

2007 ◽  
Vol 97 (6) ◽  
pp. 999-1009 ◽  
Author(s):  
Joseph Lauer ◽  
Nicholas Wormald
Keyword(s):  
Independent Sets ◽  
Regular Graphs ◽  
Large Girth

scholarly journals Computing independent sets in graphs with large girth

1992 ◽  
Vol 35 (2) ◽  
pp. 167-170 ◽  
Author(s):  
Owen J. Murphy
Keyword(s):  
Independent Sets ◽  
Large Girth

scholarly journals Minimizing the number of independent sets in triangle-free regular graphs

2018 ◽  
Vol 341 (3) ◽  
pp. 793-800 ◽  
Author(s):  
Jonathan Cutler ◽  
A.J. Radcliffe
Keyword(s):  
Independent Sets ◽  
Regular Graphs

scholarly journals Large 2-Independent Sets of Regular Graphs

2003 ◽  
Vol 78 ◽  
pp. 223-235 ◽  
Author(s):  
W. Duckworth ◽  
M. Zito
Keyword(s):  
Independent Sets ◽  
Regular Graphs

On the bipartite density of regular graphs with large girth

1990 ◽  
Vol 14 (6) ◽  
pp. 631-634 ◽  
Author(s):  
Ondřej Zýka
Keyword(s):  
Regular Graphs ◽  
Large Girth

scholarly journals Independent sets in {claw,K4}-free 4-regular graphs

2014 ◽  
Vol 332 ◽  
pp. 40-44 ◽  
Author(s):  
Liying Kang ◽  
Dingguo Wang ◽  
Erfang Shan
Keyword(s):  
Independent Sets ◽  
Regular 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.

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