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

2011
*
arXiv
*
pre-print

We study Markov chains for

arXiv:0706.1530v2
fatcat:srcsqfvllfffhexi3hecudifge
*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*. ...##
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Scheduling algorithms for multihop radio networks

1993
*
IEEE/ACM Transactions on Networking
*

Previous algorithms could guarantee only that

doi:10.1109/90.222924
fatcat:iicgyujtczbehgrjhzhfrqypqe
*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*...##
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Scheduling algorithms for multi-hop radio networks

1992
*
Computer communication review
*

Previous algorithms could guarantee only that

doi:10.1145/144191.144283
fatcat:5rb6ilo6brfu3ige654tqhl73e
*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*...##
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Scheduling algorithms for multi-hop radio networks

1992
*
Conference proceedings on Communications architectures & protocols - SIGCOMM '92
*

Previous algorithms could guarantee only that

doi:10.1145/144179.144283
dblp:conf/sigcomm/RamanathanL92
fatcat:gcld74q4urajrfewsogbdthnay
*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*...##
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D-resolvability of vertices in planar graphs

2017
*
Journal of Graph Algorithms and Applications
*

To investigate

doi:10.7155/jgaa.00433
fatcat:7ysggeje2rdwfdtkh6ys6coezq
*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. ...##
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Randomly coloring sparse random graphs with fewer colors than the maximum degree

2006
*
Random structures & algorithms (Print)
*

We show that,

doi:10.1002/rsa.20129
fatcat:gjmfgxyd7rexfgm2bmn3haru5u
*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**than*∆*colors*. ...##
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On acyclic colorings of graphs on surfaces

1996
*
Israel Journal of Mathematics
*

A proper k-

doi:10.1007/bf02762708
fatcat:joib2strubcwfetezhruoli5p4
*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 . ...##
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Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

2005
*
Journal of Discrete Algorithms
*

We study

doi:10.1016/j.jda.2004.05.002
fatcat:qo6heuw52jckhdeu2h2drt3zli
*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. ...##
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Runtime analysis of randomized search heuristics for dynamic graph coloring

2019
*
Proceedings of the Genetic and Evolutionary Computation Conference on - GECCO '19
*

We identify classes of bipartite

doi:10.1145/3321707.3321792
dblp:conf/gecco/Bossek0PS19
fatcat:qqlodxmwzncmxiyue6ef3ni4ta
*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. ...##
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Maximum Clique Conformance Measure for Graph Coloring Algorithms

2015
*
International Journal of Computer Applications
*

CCI measures

doi:10.5120/ijca2015906540
fatcat:h3qofxyvlvcc3nwaoql3bzf2zy
*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*. ...##
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How false is Kempe's proof of the Four Color Theorem? Part II

2009
*
Involve. A Journal of Mathematics
*

Kempe's "proof" gives rise to an algorithmic method of

doi:10.2140/involve.2009.2.249
fatcat:5v7rxvrtkfgdlgl6e25luxjpq4
*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 ...##
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Employing (1 − ∊) Dominating Set Partitions as Backbones in Wireless Sensor Networks
[chapter]

2010
*
2010 Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments (ALENEX)
*

r and 2r and further to form a

doi:10.1137/1.9781611972900.10
dblp:conf/alenex/MahjoubM10
fatcat:plnxqwghbjblrg5be4opv5wusa
*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. ...##
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Subexponential mixing for partition chains on grid-like graphs
[article]

2022
*
arXiv
*
pre-print

More generally, this gives sub-exponential algorithms for bounded-

arXiv:2206.00579v1
fatcat:yabvxkjka5colfgw5xd66dmqoa
*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 • ∆. ...##
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Experimental results on quadrangulations of sets of fixed points

2002
*
Computer Aided Geometric Design
*

on

doi:10.1016/s0167-8396(02)00133-4
fatcat:cisaq4vpjzhg5dcocuv6psbs64
*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. ...##
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Exploring Complex Drawings via Edge Stratification
[chapter]

2013
*
Lecture Notes in Computer Science
*

Our approach is based on stratifying

doi:10.1007/978-3-319-03841-4_27
fatcat:ry2ovxlzcbhapp54qnonw3q2gm
*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] ...
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