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Minimal coloring and strength of graphs

H. Hajiabolhassan, M.L. Mehrabadi, R. Tusserkani
2000 Discrete Mathematics  
Let G be a graph. A minimal coloring of G is a coloring which has the smallest possible sum among all proper colorings of G, using natural numbers.  ...  Acknowledgements The authors are indebted to the Research Council of the Sharif University of Technology and the Institute for Studies in Theoretical Physics and Mathematics for their support.  ...  Otherwise by deleting one of the vertices with color +1, say u, and using the minimal property of G, one can recolor every component of the graph G − u by colors to get a minimal coloring for it and obtain  ... 
doi:10.1016/s0012-365x(99)00319-2 fatcat:7xaleo5snzbrngzsi75dvtmvzy

Vertex-strength of fuzzy graphs

Changiz Eslahchi, B. N. Onagh
2006 International Journal of Mathematics and Mathematical Sciences  
The fuzzy coloring of a fuzzy graph was defined by the authors in Eslahchi and Onagh (2004). In this paper we define the chromatic fuzzy sum and strength of fuzzy graph.  ...  Some properties of these concepts are studied. It is shown that there exists an upper (a lower) bound for the chromatic fuzzy sum of a fuzzy graph.  ...  Let G be a fuzzy graph and Γ 0 = {γ 1 ,...,γ s } a minimal fuzzy sum coloring of G. Then the following results are true. (G) = 66 and Γ is not a minimal fuzzy sum coloring of G.Definition 2.7.  ... 
doi:10.1155/ijmms/2006/43614 fatcat:hsm2jea25vai7jdu4ighcsav2e

Tabular graphs and chromatic sum

H. Hajiabolhassan, M.L. Mehrabadi, R. Tusserkani
2005 Discrete Mathematics  
The chromatic sum of a graph is the minimum total of the colors on the vertices taken over all possible proper colorings using positive integers.  ...  In this paper we give some lower bounds for P (k, t) and considerably improve the upper bounds by introducing a class of graphs, called tabular graphs.  ...  Also, they are indebted to the Institute for Studies in Theoretical Physics and Mathematics and the Research Council of the Sharif University of Technology for their support.  ... 
doi:10.1016/j.disc.2005.04.022 fatcat:cgtyv3rufneh3ascbw4dbxwshy

Page 7616 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews  
(IR-SHAR; Tehran) Minimal coloring and strength of graphs. (English summary) Discrete Math. 215 (2000), no. 1-3, 265-270. Summary: “Let G be a graph.  ...  The vertex-strength of G, denoted by s(G), is the minimum number of colors which is necessary to obtain a minimal coloring.  ... 

Sum coloring and interval graphs: a tight upper bound for the minimum number of colors

S Nicoloso
2004 Discrete Mathematics  
The SUM COLORING problem consists of assigning a color c(vi) ∈ Z + to each vertex vi ∈ V of a graph G = (V; E) so that adjacent nodes have di erent colors and the sum of the c(vi)'s over all vertices vi  ...  ∈ V is minimized.  ...  Acknowledgements The author wish to thank the two anonymous referees for the helpful and accurate comments.  ... 
doi:10.1016/j.disc.2003.06.015 fatcat:tjuxhw37a5adzn7dlea7xc3tlm

On sum coloring of graphs

Mohammad R Salavatipour
2003 Discrete Applied Mathematics  
The edge sum coloring problem and the edge strength of a graph are deÿned similarly.  ...  A coloring which achieves this total sum is called an optimum coloring and the minimum number of colors needed in any optimum coloring of a graph is called the strength of the graph.  ...  Acknowledgements This was part of my M.Sc. thesis and I would like to thank my supervisor, Derek Corneil, and Mike Molloy for their helpful advice and suggestions.  ... 
doi:10.1016/s0166-218x(02)00249-4 fatcat:vq6sm3r7rvecrgm3eypn3afv5e

Collaborative Learning for Constraint Solving [chapter]

Susan L. Epstein, Eugene C. Freuder
2001 Lecture Notes in Computer Science  
It applies FORR, an architecture for learning and problemsolving, to constraint solving. FORR develops expertise from multiple heuristics. A successful case study is presented on coloring problems.  ...  The project described here seeks to automate both the application of constraint programming expertise and the extraction of domain-specific expertise.  ...  Acknowledgements This work was supported in part by NSF grant IIS-9907385 and by NASA. We thank Richard Wallace for his assistance in generating test problems.  ... 
doi:10.1007/3-540-45578-7_4 fatcat:qdwfx7ltovddblev3kbvkk5dmu

The chromatic sum of a graph: history and recent developments

Ewa Kubicka
2004 International Journal of Mathematics and Mathematical Sciences  
The strength of a graph is the minimum number of colors necessary to obtain its chromatic sum.  ...  Existing results about chromatic sum, strength of a graph, and OCCP problem are presented together with some recent developments.  ...  Jiang and West [7] provided a different construction for trees of a given strength k, in which they minimized the maximal degree rather than order. Theorem 3.2 [7] .  ... 
doi:10.1155/s0161171204306216 fatcat:sszoqjoienh5tnzxucompvcoqy

The complexity of chromatic strength and chromatic edge strength

Dániel Marx
2006 Computational Complexity  
As a first step of the proof, we present graphs for every r ≥ 3 with chromatic index r and edge strength r + 1. For some values of r, such graphs were not known before.  ...  We also study the complexity of the edge coloring version of the problem, with analogous definitions for the edge sum Σ (G) and the chromatic edge strength s (G).  ...  In particular, they pointed out an error in the gadget construction of Theorem 2.2, and simplified the proof of Theorem 5.2.  ... 
doi:10.1007/s00037-005-0201-2 fatcat:3efxub3nkzcgvonyq255w7ztja

Geodesic graph cut for interactive image segmentation

Brian L. Price, Bryan Morse, Scott Cohen
2010 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition  
Rather than a fixed combination we use the distinctiveness of the foreground/background color models to predict the effectiveness of the geodesic distance term and adjust the weighting accordingly.  ...  Methods that grow regions from foreground/background seeds, such as the recent geodesic segmentation approach, avoid the boundary-length bias of graph-cut methods but have their own bias towards minimizing  ...  color models to automatically tune the tradeoff between the strengths and weaknesses of the two.  ... 
doi:10.1109/cvpr.2010.5540079 dblp:conf/cvpr/PriceMC10a fatcat:n7ioq552ozghlkucqccggx73xe

Page 4625 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
H(G) is defined as the maximum cardinality of a minimal harmonious coloring of a graph G, while H’(G) is defined as the maximum cardinality of a minimal line-distinguishing coloring of a graph G.  ...  The strength is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree with strength k that has maximum degree only 2k — 2.  ... 

Minimum sum edge colorings of multicycles

Jean Cardinal, Vlady Ravelomanana, Mario Valencia-Pabon
2010 Discrete Applied Mathematics  
The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph.  ...  In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum  ...  Acknowledgments We thank Samuel Fiorini for insightful discussions on this topic, and the anonymous referees for their helpful comments.  ... 
doi:10.1016/j.dam.2009.04.020 fatcat:ptau5ewmmjakbmhifmy2j6zvfy

Page 654 of Mathematical Reviews Vol. , Issue 91B [page]

1991 Mathematical Reviews  
The acyclic dichromatic number of a digraph is the minimum number of colors required to color its vertices in such a way that every color class induces an acyclic subdigraph in it.  ...  Given that G has order n, strength m, and maximum degree d, the smallest nonneg- ative integer r for which there exists a d-regular supermultigraph of G with strength m and order n+r is called the m-regulation  ... 

Page 9587 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
The chromatic number of H is the minimal & for which H admits a k-coloring. This paper fills a gap between the well-known case of graphs (r = 2) and the case of a general r.  ...  classes: interval graphs, circular-arc graphs and m-trapezoid graphs (co-comparability graphs of interval dimension m+ | or- ders).  ... 

A hybrid algorithm for the sum coloring problem

Sidi Mohamed Douiri, Souad El Bernoussi
2011 2011 International Conference on Multimedia Computing and Systems  
The problem (MSCP) consists in minimizing the sum of colors in a graph.  ...  In this paper we are interested in the elaboration of an approached solution to the sum coloring problem (MSCP), which is an NP-hard problem derived from the graphs coloring (GCP).  ...  Description of the approach to the resolution of (MSCP) Our algorithm is applied to find both the sum of minimum graph coloring (G) and the strength s(G).  ... 
doi:10.1109/icmcs.2011.5945684 fatcat:3zbaz4n2pvdk5byg22cb64klja
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