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Minimal coloring and strength of graphs

2000
*
Discrete Mathematics
*

Let G be a

doi:10.1016/s0012-365x(99)00319-2
fatcat:7xaleo5snzbrngzsi75dvtmvzy
*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 ...##
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Vertex-strength of fuzzy graphs

2006
*
International Journal of Mathematics and Mathematical Sciences
*

The fuzzy

doi:10.1155/ijmms/2006/43614
fatcat:hsm2jea25vai7jdu4ighcsav2e
*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. ...##
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Tabular graphs and chromatic sum

2005
*
Discrete Mathematics
*

The chromatic sum

doi:10.1016/j.disc.2005.04.022
fatcat:cgtyv3rufneh3ascbw4dbxwshy
*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. ...##
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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*. ...##
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Sum coloring and interval graphs: a tight upper bound for the minimum number of colors

2004
*
Discrete Mathematics
*

The SUM

doi:10.1016/j.disc.2003.06.015
fatcat:tjuxhw37a5adzn7dlea7xc3tlm
*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. ...##
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On sum coloring of graphs

2003
*
Discrete Applied Mathematics
*

The edge sum

doi:10.1016/s0166-218x(02)00249-4
fatcat:vq6sm3r7rvecrgm3eypn3afv5e
*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. ...##
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Collaborative Learning for Constraint Solving
[chapter]

2001
*
Lecture Notes in Computer Science
*

It applies FORR, an architecture for learning

doi:10.1007/3-540-45578-7_4
fatcat:qdwfx7ltovddblev3kbvkk5dmu
*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. ...##
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The chromatic sum of a graph: history and recent developments

2004
*
International Journal of Mathematics and Mathematical Sciences
*

The

doi:10.1155/s0161171204306216
fatcat:sszoqjoienh5tnzxucompvcoqy
*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] . ...##
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The complexity of chromatic strength and chromatic edge strength

2006
*
Computational Complexity
*

As a first step

doi:10.1007/s00037-005-0201-2
fatcat:3efxub3nkzcgvonyq255w7ztja
*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. ...##
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Geodesic graph cut for interactive image segmentation

2010
*
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition
*

Rather than a fixed combination we use the distinctiveness

doi:10.1109/cvpr.2010.5540079
dblp:conf/cvpr/PriceMC10a
fatcat:n7ioq552ozghlkucqccggx73xe
*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. ...##
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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. ...##
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Minimum sum edge colorings of multicycles

2010
*
Discrete Applied Mathematics
*

The chromatic edge

doi:10.1016/j.dam.2009.04.020
fatcat:ptau5ewmmjakbmhifmy2j6zvfy
*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. ...##
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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 ...##
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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). ...##
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A hybrid algorithm for the sum coloring problem

2011
*
2011 International Conference on Multimedia Computing and Systems
*

The problem (MSCP) consists in

doi:10.1109/icmcs.2011.5945684
fatcat:3zbaz4n2pvdk5byg22cb64klja
*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). ...
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