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.

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 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.

scholarly journals Size Ramsey Number of Bounded Degree Graphs for Games

2013 ◽  
Vol 22 (4) ◽  
pp. 499-516
Author(s):  
HEIDI GEBAUER
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Minimum Number ◽  
Game Theoretic ◽  
Bounded Degree Graphs ◽  
Constant Maximum

We study Maker/Breaker games on the edges ofsparsegraphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graphH. Maker aims to occupy a given target graphGand Breaker tries to prevent Maker from achieving his goal. We show that for everydthere is a constantc=c(d)with the property that for every graphGonnvertices of maximum degreedthere is a graphHon at mostcnedges such that Maker has a strategy to occupy a copy ofGin the game onH.This is a result about a game-theoretic variant of the size Ramsey number. For a given graphG,$\hat{r}'(G)$is defined as the smallest numberMfor which there exists a graphHwithMedges such that Maker has a strategy to occupy a copy ofGin the game onH. In this language, our result yields that for every connected graphGof constant maximum degree,$\hat{r}'(G) = \Theta(n)$.Moreover, we can also use our method to settle the corresponding extremal number foruniversalgraphs: for a constantdand for the class${\cal G}_{n}$ofn-vertex graphs of maximum degreed,$s({\cal G}_{n})$denotes the minimum number such that there exists a graphHwithMedges where, foreveryG∈${\cal G}_{n}$, Maker has a strategy to build a copy ofGin the game onH. We obtain that$s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})$.

Anagram-Free Colourings of Graphs

2017 ◽  
Vol 27 (4) ◽  
pp. 623-642 ◽  
Author(s):  
NINA KAMČEV ◽  
TOMASZ ŁUCZAK ◽  
BENNY SUDAKOV
Keyword(s):  
Chromatic Number ◽  
Natural Generalization ◽  
Regular Graphs ◽  
Bounded Degree ◽  
Bounded Degree Graphs ◽  
Graph Colourings ◽  
Minor Free Graphs

A sequenceSis calledanagram-freeif it contains no consecutive symbolsr1r2. . .rkrk+1. . .r2ksuch thatrk+1. . .r2kis a permutation of the blockr1r2. . .rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graphGis calledanagram-freeif the sequence of colours on any path inGis anagram-free. We call the minimal number of colours needed for such a colouring theanagram-chromaticnumber ofG.In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we essentially give each vertex a separate colour.

scholarly journals Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

2021 ◽  
Author(s):  
Antonio Blanca ◽  
Reza Gheissari
Keyword(s):  
Phase Transition ◽  
Potts Model ◽  
Cluster Model ◽  
Iterative Scheme ◽  
Mixing Time ◽  
Glauber Dynamics ◽  
Regular Graphs ◽  
Random Cluster Model ◽  
Regular Tree ◽  
Random Cluster

AbstractWe establish rapid mixing of the random-cluster Glauber dynamics on random $$\varDelta $$ Δ -regular graphs for all $$q\ge 1$$ q ≥ 1 and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , where the threshold $$p_u(q,\varDelta )$$ p u ( q , Δ ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $$\varDelta $$ Δ -regular tree. It is expected that this threshold is sharp, and for $$q>2$$ q > 2 the Glauber dynamics on random $$\varDelta $$ Δ -regular graphs undergoes an exponential slowdown at $$p_u(q,\varDelta )$$ p u ( q , Δ ) . More precisely, we show that for every $$q\ge 1$$ q ≥ 1 , $$\varDelta \ge 3$$ Δ ≥ 3 , and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , with probability $$1-o(1)$$ 1 - o ( 1 ) over the choice of a random $$\varDelta $$ Δ -regular graph on n vertices, the Glauber dynamics for the random-cluster model has $$\varTheta (n \log n)$$ Θ ( n log n ) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random $$\varDelta $$ Δ -regular graphs for every $$q\ge 2$$ q ≥ 2 , in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into $$O(\log n)$$ O ( log n ) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.

scholarly journals Large Bounded Degree Trees in Expanding Graphs

10.37236/278 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
József Balogh ◽  
Béla Csaba ◽  
Martin Pei ◽  
Wojciech Samotij
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Spectral Gap ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Bounded Degree ◽  
Large Trees ◽  
Remarkable Result ◽  
Bounded Degree Trees ◽  
Expanding Graphs

A remarkable result of Friedman and Pippenger gives a sufficient condition on the expansion properties of a graph to contain all small trees with bounded maximum degree. Haxell showed that under slightly stronger assumptions on the expansion rate, their technique allows one to find arbitrarily large trees with bounded maximum degree. Using a slightly weaker version of Haxell's result we prove that a certain family of expanding graphs, which includes very sparse random graphs and regular graphs with large enough spectral gap, contains all almost spanning bounded degree trees. This improves two recent tree-embedding results of Alon, Krivelevich and Sudakov.

scholarly journals Cycle transversals in bounded degree graphs

2011 ◽  
Vol Vol. 13 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Marina Groshaus ◽  
Pavol Hell ◽  
Sulamita Klein ◽  
Loana Tito Nogueira ◽  
Fábio Protti
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Decomposition Theorem ◽  
Np Hard ◽  
Input Graph ◽  
Bounded Degree ◽  
Time Approximation ◽  
International Audience ◽  
Bounded Degree Graphs

Graphs and Algorithms International audience In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.

scholarly journals Degree Ramsey Numbers of Graphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 229-253 ◽  
Author(s):  
WILLIAM B. KINNERSLEY ◽  
KEVIN G. MILANS ◽  
DOUGLAS B. WEST
Keyword(s):  
Maximum Degree ◽  
Double Star ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Ramsey Numbers ◽  
Colour Class ◽  
Bounded Degree Graphs ◽  
Edge Colourings ◽  
Edge Colouring ◽  
Monochromatic Copy

Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

scholarly journals Testing Expansion in Bounded-Degree Graphs

2010 ◽  
Vol 19 (5-6) ◽  
pp. 693-709 ◽  
Author(s):  
ARTUR CZUMAJ ◽  
CHRISTIAN SOHLER
Keyword(s):  
Maximum Degree ◽  
Property Testing ◽  
Graph Representation ◽  
Bounded Degree ◽  
Running Time ◽  
Adjacency List ◽  
Bounded Degree Graphs ◽  
Testing Algorithm

We consider the problem of testing expansion in bounded-degree graphs. We focus on the notion of vertex expansion: an α-expander is a graph G = (V, E) in which every subset U ⊆ V of at most |V|/2 vertices has a neighbourhood of size at least α ⋅ |U|. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time $\widetilde{\O}(\sqrt{n})$. We prove that the property-testing algorithm proposed by Goldreich and Ron with appropriately set parameters accepts every α-expander with probability at least $\frac23$ and rejects every graph that is ϵ-far from any α*-expander with probability at least $\frac23$, where $\expand^* \,{=}\, \Theta(\frac{\expand^2}{d^2 \log(n/\epsilon)})$ and d is the maximum degree of the graphs. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is $\O(\frac{d^2 \sqrt{n} \log(n/\epsilon)} {\expand^2 \epsilon^3})$.

scholarly journals A Note on the Glauber Dynamics for Sampling Independent Sets

10.37236/1552 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Eric Vigoda
Keyword(s):  
Maximum Degree ◽  
Mixing Time ◽  
Independent Set ◽  
Glauber Dynamics ◽  
Independent Sets ◽  
General Graphs

This note considers the problem of sampling from the set of weighted independent sets of a graph with maximum degree $\Delta$. For a positive fugacity $\lambda$, the weight of an independent set $\sigma$ is $\lambda^{|\sigma|}$. Luby and Vigoda proved that the Glauber dynamics, which only changes the configuration at a randomly chosen vertex in each step, has mixing time $O(n\log{n})$ when $\lambda < {{2}\over {\Delta-2}}$ for triangle-free graphs. We extend their approach to general graphs.

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