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 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 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.

scholarly journals The analysis of a prioritised probabilistic algorithm to find large induced forests in regular graphs with large girth

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Carlos Hoppen
Keyword(s):  
Lower Bounds ◽  
Regular Graphs ◽  
Probabilistic Algorithms ◽  
Induced Subgraph ◽  
Fixed Integer ◽  
Asymptotic Bounds ◽  
Random Regular Graph ◽  
International Audience ◽  
Large Girth ◽  
Deterministic Bounds

International audience The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.

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