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Randomly coloring planar graphs with fewer colors than the maximum degree [article]

Thomas P. Hayes, Juan C. Vera, Eric Vigoda
2011 arXiv   pre-print
We study Markov chains for randomly sampling k-colorings of a graph with maximum degree Δ.  ...  Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when k=Ω(Δ/Δ).  ...  We now point out that, for graphs with small operator norm or, equivalently, small δ, the Glauber dynamics is ergodic with many fewer colors than the maximum degree.  ... 
arXiv:0706.1530v2 fatcat:srcsqfvllfffhexi3hecudifge

Scheduling algorithms for multihop radio networks

S. Ramanathan, E.L. Lloyd
1993 IEEE/ACM Transactions on Networking  
Previous algorithms could guarantee only that the schedules were bounded by a length no worse than the maximum node degree times optimum.  ...  Since the thickness is typically several orders of magnitude less than the maximum node degree, the algorithms presented here represent a considerable theoretical improvement.  ...  oriented graphs, 0 is the maximum number of neighbors with lower labels, and p is the maximum vertex degree. 3) Planar networks: Consider a call of the function labeler when the input graph G is planar  ... 
doi:10.1109/90.222924 fatcat:iicgyujtczbehgrjhzhfrqypqe

Scheduling algorithms for multi-hop radio networks

S. Ramanathan, Errol L. Lloyd
1992 Computer communication review  
Previous algorithms could guarantee only that the schedules were bounded by a length no worse than the maximum node degree times optimum.  ...  Since the thickness is typically several orders of magnitude less than the maximum node degree, the algorithms presented here represent a considerable theoretical improvement.  ...  oriented graphs, 0 is the maximum number of neighbors with lower labels, and p is the maximum vertex degree. 3) Planar networks: Consider a call of the function labeler when the input graph G is planar  ... 
doi:10.1145/144191.144283 fatcat:5rb6ilo6brfu3ige654tqhl73e

Scheduling algorithms for multi-hop radio networks

S. Ramanathan, Errol L. Lloyd
1992 Conference proceedings on Communications architectures & protocols - SIGCOMM '92  
Previous algorithms could guarantee only that the schedules were bounded by a length no worse than the maximum node degree times optimum.  ...  Since the thickness is typically several orders of magnitude less than the maximum node degree, the algorithms presented here represent a considerable theoretical improvement.  ...  oriented graphs, 0 is the maximum number of neighbors with lower labels, and p is the maximum vertex degree. 3) Planar networks: Consider a call of the function labeler when the input graph G is planar  ... 
doi:10.1145/144179.144283 dblp:conf/sigcomm/RamanathanL92 fatcat:gcld74q4urajrfewsogbdthnay

D-resolvability of vertices in planar graphs

James A. Tilley
2017 Journal of Graph Algorithms and Applications  
To investigate the question, we designed and implemented two algorithms, one to 4-color a planar graph and the other to generate internally 6-connected triangulations randomly.  ...  It is well known that a vertex of degree 5 in a planar graph is not Dreducible. However, it remains an open question whether all vertices in such a graph are D-resolvable.  ...  This is a very weak lower bound because we know that two planar graphs with identical sets of vertex degrees can be non-isomorphic by reason of differing adjacencies.  ... 
doi:10.7155/jgaa.00433 fatcat:7ysggeje2rdwfdtkh6ys6coezq

Randomly coloring sparse random graphs with fewer colors than the maximum degree

Martin Dyer, Abraham D. Flaxman, Alan M. Frieze, Eric Vigoda
2006 Random structures & algorithms (Print)  
We show that, with high probability, an efficient procedure can generate an almost uniformly random k-coloring when k = Θ(log log n/ log log log n), i.e., with many fewer colors than the maximum degree  ...  Previous results hold for a more general class of graphs, but always require more colors than the maximum degree.  ...  We randomly color such graphs with many fewer thancolors.  ... 
doi:10.1002/rsa.20129 fatcat:gjmfgxyd7rexfgm2bmn3haru5u

On acyclic colorings of graphs on surfaces

Noga Alon, Bojan Mohar, Daniel P. Sanders
1996 Israel Journal of Mathematics  
A proper k-coloring of a graph is acyclic if every 2-chromatic subgraph is acyclic. Borodin showed that every planar graph has an acyclic 5-coloring.  ...  Therefore, the conjecture of Borodin that the acyclic chromatic number of any surface but the plane is the same as its chromatic number is false for all surfaces with large γ (and may very well be false  ...  The proof uses the following result from [4] . Lemma 3 (Alon, McDiarmid, Reed) . For every graph G with maximum degree d, A(G) ≤ 50d 4/3 .  ... 
doi:10.1007/bf02762708 fatcat:joib2strubcwfetezhruoli5p4

Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

Tomasz Łuczak, Eric Vigoda
2005 Journal of Discrete Algorithms  
We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree ∆.  ...  In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3 k < ∆/(20 log ∆), and on a family of planar graphs for any number of colors.  ...  the manuscript.  ... 
doi:10.1016/j.jda.2004.05.002 fatcat:qo6heuw52jckhdeu2h2drt3zli

Runtime analysis of randomized search heuristics for dynamic graph coloring

Jakob Bossek, Frank Neumann, Pan Peng, Dirk Sudholt
2019 Proceedings of the Genetic and Evolutionary Computation Conference on - GECCO '19  
We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i. e. starting with a random initialization.  ...  We consider the classical graph coloring problem and investigate the dynamic se ing where edges are added to the current graph.  ...  Bossek acknowledges support by the European Research Center for Information Systems (ERCIS). F. Neumann has been supported by the Australian Research Council (ARC) through grant DP160102401.  ... 
doi:10.1145/3321707.3321792 dblp:conf/gecco/Bossek0PS19 fatcat:qqlodxmwzncmxiyue6ef3ni4ta

Maximum Clique Conformance Measure for Graph Coloring Algorithms

Abdulmutaleb Alzubi, Mohammad Al-Haj, Mohammad Malkawi
2015 International Journal of Computer Applications  
CCI measures the rate of deviation of a coloring algorithm from the maximum clique during the process of coloring a graph.  ...  This paper presents a new measure for evaluating the efficiency of graph coloring algorithms. The new measure computes the clique conformance index (CCI) of a graph coloring algorithm.  ...  An interesting coloring problem is the four colors theorem, which colors certain planar graphs with exactly four colors.  ... 
doi:10.5120/ijca2015906540 fatcat:h3qofxyvlvcc3nwaoql3bzf2zy

How false is Kempe's proof of the Four Color Theorem? Part II

Gethner Ellen, Bopanna Kallichanda, Alexander Mentis, Sarah Braudrick, Sumeet Chawla, Andrew Clune, Rachel Drummond, Panagiota Evans, William Roche, Nao Takano
2009 Involve. A Journal of Mathematics  
Kempe's "proof" gives rise to an algorithmic method of coloring plane graphs that sometimes yields a proper vertex coloring requiring four or fewer colors.  ...  Then we empirically analyze the performance of the implementations on a variety of historically motivated benchmark graphs and explore the usefulness of simple randomization in four-coloring small plane  ...  In 500 test runs on each of ten benchmark graphs, the use of randomized chain choices resulted in successful four-coloring of the graphs with fewer than 12 random choices for any vertex most of the time  ... 
doi:10.2140/involve.2009.2.249 fatcat:5v7rxvrtkfgdlgl6e25luxjpq4

Employing (1 − ∊) Dominating Set Partitions as Backbones in Wireless Sensor Networks [chapter]

Dhia Mahjoub, David W. Matula
2010 2010 Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments (ALENEX)  
r and 2r and further to form a planar graph backbone.  ...  The backbone node sets are determined by a graph coloring algorithm employing only the topology (not the geometry) of G(n, r), and the backbone links are selected with link lengths in a narrow window between  ...  Acknowledgment The authors would like to thank Dr.  ... 
doi:10.1137/1.9781611972900.10 dblp:conf/alenex/MahjoubM10 fatcat:plnxqwghbjblrg5be4opv5wusa

Subexponential mixing for partition chains on grid-like graphs [article]

Alan Frieze, Wesley Pegden
2022 arXiv   pre-print
More generally, this gives sub-exponential algorithms for bounded-degree graphs without large expander subgraphs (for example, we obtain O(2^√(n)) time algorithms for square grids).  ...  We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph.  ...  Define the graph H whose vertex-set is the color classes of c, with two color classes being adjacent in H if they are joined by any edge in G. Note that the maximum degree of H is at most B • ∆.  ... 
arXiv:2206.00579v1 fatcat:yabvxkjka5colfgw5xd66dmqoa

Experimental results on quadrangulations of sets of fixed points

Prosenjit Bose, Suneeta Ramaswami, Godfried Toussaint, Alain Turki
2002 Computer Aided Geometric Design  
on the dual graph of the triangulations.  ...  Therefore we propose and experimentally investigate a heuristic approach to solve this problem by converting "nice" triangulations to the desired quadrangulations with the aid of maximum matchings computed  ...  This triangulation also has the worst results for the average angle measures, with the maximum angle being greater than 189 degrees, and the minimum angle being less than 23 degrees on average.  ... 
doi:10.1016/s0167-8396(02)00133-4 fatcat:cisaq4vpjzhg5dcocuv6psbs64

Exploring Complex Drawings via Edge Stratification [chapter]

Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, Ioannis G. Tollis
2013 Lecture Notes in Computer Science  
Our approach is based on stratifying the drawing into layers with desired properties; layers can be explored and combined by the user to gradually acquire details.  ...  We propose an approach that allows a user to explore a layout produced by any graph drawing algorithm, in order to reduce the visual complexity and clarify its presentation.  ...  We prove that they are difficult at least as the well-known problem called classification, restricted to planar graphs with maximum vertex degree 4 or 5; this problem is conjectured to be NP-hard [9]  ... 
doi:10.1007/978-3-319-03841-4_27 fatcat:ry2ovxlzcbhapp54qnonw3q2gm
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