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Acyclic and k-distance coloring of the grid

Guillaume Fertin, Emmanuel Godard, André Raspaud
2003 Information Processing Letters  
¦ @ ¼ j 6 H Q P 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 )  ... 
doi:10.1016/s0020-0190(03)00232-1 fatcat:twbage3fqfbzthyx2butsz5g7u

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

Guillaume Fertin, Andr� Raspaud, Bruce Reed
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.  ... 
doi:10.1002/jgt.20029 fatcat:cx6yostynjhnxdfhoazracwxyu

On the Acyclic Chromatic Number of Hamming Graphs

Robert E. Jamison, Gretchen L. Matthews
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  ... 
doi:10.1007/s00373-008-0798-4 fatcat:aoo5qggdy5fkxpbg7vel7p6224

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

Andreas Potthoff, Sebastian Seibert, Wolfgang Thomas
1994 Bulletin of the Belgian Mathematical Society Simon Stevin  
Three types 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 ©  ... 
doi:10.36045/bbms/1103408550 fatcat:b6vwobes35fitmxrikut4gj7q4

Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

Georgia Kouyialis, Xiaoyu Wang, Ruth Misener
2019 Processes  
This paper highlights 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  ... 
doi:10.3390/pr7110838 fatcat:s2uuuxa73vggjj3kn7d4w4t33q

Compact Grid Representation of Graphs [chapter]

José Cáceres, Carmen Cortés, Clara Isabel Grima, Masahiro Hachimori, Alberto Márquez, Raiji Mukae, Atsuhiro Nakamoto, Seiya Negami, Rafael Robles, Jesús Valenzuela
2012 Lecture Notes in Computer Science  
Additionally G is said to be properly embeddable in 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  ... 
doi:10.1007/978-3-642-34191-5_16 fatcat:tpdtata7w5eerh4nyics6zzb7e

Dynamic monopolies and feedback vertex sets in hexagonal grids

Sarah Spence Adams, Denise Sakai Troxell, S. Luke Zinnen
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  ... 
doi:10.1016/j.camwa.2011.09.047 fatcat:3wn4cak3zrfqrcifjyqq2oe5b4

Acyclic coloring of graphs

Noga Alon, Colin Mcdiarmid, Bruce Reed
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.  ... 
doi:10.1002/rsa.3240020303 fatcat:dq3ainuygfcrnajo2si56jnada

Fast convergence in self-stabilizing wireless networks

N. Mitton, E. Fleury, I.G. Lassous, B. Sericola, S. Tixeuil
2006 12th International Conference on Parallel and Distributed Systems - (ICPADS'06)  
Similarly, 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  ... 
doi:10.1109/icpads.2006.47 dblp:conf/icpads/MittonFLST06 fatcat:acdghqqvtneovg7w33npnu6wyq

Acyclic Coloring of Graphs of Maximum Degree $\Delta$

Guillaume Fertin, André Raspaud
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) .  ... 
doi:10.46298/dmtcs.3450 fatcat:jldhrresqbc6fepwg5skyjs6rq

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

Guillaume Bagan and Alice Joffard and Hamamache Kheddouci
2018 arXiv   pre-print
We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs 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.  ... 
arXiv:1805.09623v1 fatcat:mgctg2j2uzci7pxwqf4juv444u

DAGView: An Approach for Visualizing Large Graphs [chapter]

Evgenios M. Kornaropoulos, Ioannis G. Tollis
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.  ... 
doi:10.1007/978-3-642-36763-2_44 fatcat:jyxjxhcemzferos2xgep37lwga

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

Zongqing Chen, Sergey Kitaev, Brian Y. Sun
2015 arXiv   pre-print
A subdivision 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.  ... 
arXiv:1503.08002v1 fatcat:di3tjglesrazhmjdrgtpvecrpa
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