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Acyclic and k-distance coloring of the grid

2003
*
Information Processing Letters
*

¦ @ ¼ j 6 H Q P

doi:10.1016/s0020-0190(03)00232-1
fatcat:twbage3fqfbzthyx2butsz5g7u
*k*q c x ¬ n B H A 2 ! ...*k*@ Y S £ ¥ ¤ p i 1 q ¦ t v w » e v y W h*k*n*k*0 £ 2 0 @ I t v w s e v y h 6 8 w x p r 6 I t 5 ( ® g ¥ t ¡ s t v p » " y 2 x p r t v y e 2 £ · r I t 5 ( v " ¥ t ® 1 ¤ q ' ¡ I t x ( 5 ( n £ ¢ s 9 % 2 ... H v C S 9 g d 9 " A ¡ B 1 ' F " 7 %*k*r @ 9 p 9 g 7 3 I 1 Ỳ a U 3 I 9 " 9 g % ' I ) ...##
###
Page 8408 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

For a graph G

*of*size m > 1*and*an integer*k*with 1 <*k*< m,*the**k*-jump graph J,(G) is that graph whose vertices corre- spond to*the*edge-induced subgraphs*of*size*k**of*G*and*where two vertices*of*J,(G) ... An*acyclic**coloring**of*a graph G is an assignment*of**colors*to*the*vertices*of*G such that each vertex has exactly one*color*, each edge has exactly two*colors*,*and*each cycle has at least three*colors*. ...##
###
Star coloring of graphs

2004
*
Journal of Graph Theory
*

*The*star chromatic number

*of*an undirected graph G, denoted by χs(G), is

*the*smallest integer

*k*for which G admits a star

*coloring*with

*k*

*colors*. ... In this paper, we give

*the*exact value

*of*

*the*star chromatic number

*of*different families

*of*graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs

*and*2-dimensional

*grids*. ...

*The*authors wish to thank

*the*anonymous referees for many valuable remarks that helped to improve this paper. ...

##
###
On the Acyclic Chromatic Number of Hamming Graphs

2008
*
Graphs and Combinatorics
*

*The*

*acyclic*chromatic number

*of*a graph G is

*the*minimum number

*k*such that G has an

*acyclic*

*coloring*with

*k*

*colors*. ... In this paper,

*acyclic*

*colorings*

*of*Hamming graphs, products

*of*complete graphs, are considered. Upper

*and*lower bounds on

*the*

*acyclic*chromatic number

*of*Hamming graphs are given. ...

*The*authors thank

*the*referee for useful comments. On

*the*

*acyclic*chromatic number

*of*Hamming graphs ...

##
###
Page 1712 of Mathematical Reviews Vol. , Issue 2004c
[page]

2004
*
Mathematical Reviews
*

2004c:05069 05
2004c:05069 0S5CIS 68R10
Fertin, Guillaume (F-NANT-IR; Nantes) ;
Godard, Emmanuel (F-BORD-LB; Talence) ;
Raspaud, André (F-BORD-LB; Talence)

*Acyclic**and**k*-*distance**coloring**of**the**grid*. ...*the*same*color**and*(ii) for any two*colors*i*and*/,*the*subgraph induced by*the*vertices*colored*i or j is*acyclic*;*and*(2)*k*-*distance**coloring**of*graphs, in which every vertex must be*colored*in such ...##
###
Nondeterminism versus determinism of finite automata over directed acyclic graphs

1994
*
Bulletin of the Belgian Mathematical Society Simon Stevin
*

Three types

doi:10.36045/bbms/1103408550
fatcat:b6vwobes35fitmxrikut4gj7q4
*of*finite-state graph automata are compared over directed*acyclic*graphs (where vertices*and*edges are labelled). ... We note*the*equivalence*of*these models, relate them to logical definability notions,*and*show that deterministic versions are strictly weaker (thus correcting an error*of*[9]). ... Consider*the*graphs G*k*= (V*k*, E*k*, µ) where V*k*= {0, . . . , 2k − 1},*and*E*k*consists*of**the*edges {(2i, 2i + 1) | i <*k*}*colored*c 2*and*{(2i, 2i − 1(mod(2k))) | 0 < i <*k*}*colored*c 1 . d d d © ...##
###
Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

2019
*
Processes
*

This paper highlights

doi:10.3390/pr7110838
fatcat:s2uuuxa73vggjj3kn7d4w4t33q
*the*importance*of*detecting*and*classifying symmetry*and*shows that knowledge*of*this symmetry enables quick approximation*of*a highly symmetric optimization problem. ... We enter*the*binary layered graphs into*the*software package nauty that generates important symmetric properties*of**the*original problem. ... Abbreviations*The*following abbreviations are used in this manuscript: MINLP Mixed-integer nonlinear optimization QCQP Quadratically-constrained quadratic optimization ...##
###
Compact Grid Representation of Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

Additionally G is said to be properly embeddable in

doi:10.1007/978-3-642-34191-5_16
fatcat:tpdtata7w5eerh4nyics6zzb7e
*the**grid*if it is*grid*locatable*and**the*segments representing edges do not cross each other. ... A graph G is said to be*grid*locatable if it admits a representation such that vertices are mapped to*grid*points*and*edges to line segments that avoid*grid*points but*the*extremes. ... Similarly, for a directed graph G, δ − (v) denotes*the*in-degree*of*a vertex v*and*Δ − (G)*the*maximum vertex in-degree in G.Lemma 1.Let G be a connected, directed,*acyclic**and*not isomorphic to*K*5 graph ...##
###
Dynamic monopolies and feedback vertex sets in hexagonal grids

2011
*
Computers and Mathematics with Applications
*

*and*toroidal hexagonal

*grids*. ... We use this equivalence to provide exact values for

*the*minimum size

*of*dynamic monopolies

*of*planar hexagonal

*grids*, as well as upper

*and*lower bounds on

*the*minimum size

*of*dynamic monopolies

*of*cylindrical ...

*The*authors would like to thank

*the*reviewers for their reports which led to significant improvements in this manuscript;

*the*authors are grateful for

*the*time

*and*care that went into preparing these reports ...

##
###
Acyclic coloring of graphs

1991
*
Random structures & algorithms (Print)
*

*The*smallest number

*of*

*colors*required to

*acyclically*vertex

*color*a graph is called its

*acyclic*chromatic number

*and*is denoted by a(G). ... Similarly,

*the*smallest number

*of*

*colors*required to

*acyclically*edge

*colors*a graph is called its

*acyclic*edge chromatic number

*and*is denoted by a ′ (G). ...

*Acyclic*Vertex

*Coloring*Fertin et al. [FGR03] have explored

*the*

*acyclic*vertex

*coloring*

*of*

*grids*which are a subclass

*of*

*the*more general d-dimensional partial tori graphs. ...

##
###
Fast convergence in self-stabilizing wireless networks

2006
*
12th International Conference on Parallel and Distributed Systems - (ICPADS'06)
*

Similarly,

doi:10.1109/icpads.2006.47
dblp:conf/icpads/MittonFLST06
fatcat:acdghqqvtneovg7w33npnu6wyq
*the*height*of**the*induced DAG is much lower than*the*linear dependency on*the*size*of**the**color*domain (that was previously announced). ...*The*advent*of*large scale multi-hop wireless networks highlights problems*of*fault tolerance*and*scale in distributed systems, motivating designs that autonomously recover from transient faults*and*spontaneous ... We have run Algorithm N1 for*distance*-1*and**distance*-2*colorings*with these scheduling hypothesis*and*modes over a random geometric topology*and*a*grid*with a varying number*of*neighbors per node*and*different ...##
###
Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005
*
Discrete Mathematics & Theoretical Computer Science
*

*The*

*acyclic*chromatic number

*of*$G$ is

*the*least number

*of*

*colors*necessary to

*acyclically*

*color*$G$,

*and*is denoted by $a(G)$. ... International audience An

*acyclic*

*coloring*

*of*a graph $G$ is a

*coloring*

*of*its vertices such that: (i) no two neighbors in $G$ are assigned

*the*same

*color*

*and*(ii) no bicolored cycle can exist in $G$. ... We also note that there exist graphs

*of*maximum degree 5 with

*acyclic*chromatic number 6, such as

*the*complete graph

*K*6 ,

*the*complete bipartite graph

*K*5,5 or

*the*circulant graph C 8 (1, 2, 4) . ...

##
###
Eternal dominating sets on digraphs and orientations of graphs
[article]

2018
*
arXiv
*
pre-print

We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs

arXiv:1805.09623v1
fatcat:mgctg2j2uzci7pxwqf4juv444u
*and*different kinds*of**grids**and*products*of*graphs. ... We study*the*eternal dominating number*and**the*m-eternal dominating number on digraphs. We generalize known results on graphs to digraphs. ... Let p be*the**color**of*v i*and*q be*the**color**of*u j . Then, we assign to w i,j*the**color*r = p + q mod*k*. Let r ′ be a*color*different from r. ...##
###
DAGView: An Approach for Visualizing Large Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

*The*aim

*of*DAGView is to produce clear visualizations

*of*directed

*acyclic*graphs in which every edge

*and*

*the*potential existence

*of*a path can be immediately spotted by

*the*user. ...

*The*main algorithm is based on

*the*layout

*of*directed

*acyclic*graphs but can be extended to handle directed graphs with cycles

*and*undirected graphs, taking into account user preferences

*and*/or constraints ... Columns denote

*the*origin-node

*of*

*the*edge, while

*the*rows denote

*the*target-node

*of*

*the*edge. If there is an edge (u, v)

*the*cell at

*the*intersection

*of*column u

*and*row v, is

*colored*black. ...

##
###
Word-representability of subdivisions of triangular grid graphs
[article]

2015
*
arXiv
*
pre-print

A subdivision

arXiv:1503.08002v1
fatcat:di3tjglesrazhmjdrgtpvecrpa
*of*a triangular*grid*graph is replacing some*of*its cells by plane copies*of**the*complete graph K_4. ... A graph G=(V,E) is word-representable if there exists a word w over*the*alphabet V such that letters x*and*y alternate in w if*and*only if (x,y)∈ E. ... Note that in our definition, a triangular*grid*graph does not have to be an induced subgraph*of*T ∞ . It is easy to see that T ∞ is 3-*colorable*,*and*thus any triangular*grid*graph is 3-*colorable*. ...
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