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Achromatic number, achromatic index and diachromatic number of circulant graphs and digraphs [article]

Gabriela Araujo-Pardo, Juan José Montellano-Ballesteros, Mika Olsen, Christian Rubio-Montiel
2019 arXiv   pre-print
Finally, we determine the achromatic index for circulant graphs of q^2+q+1 vertices when the projective cyclic plane of odd order q exists.  ...  In general, we give the lower bounds α(C_4q^2+aq+1(1,a))≥ 4q+1 and dac(C_8q^2+2(a+4)q+a+3(1,a))≥ 4q+3 when a is a non quadratic residue of Z_4q+1 for graphs and Z_4q+3 for digraphs, and the equality is  ...  On the other hand, in the collection of papers [4, 6, 7, 15, 17, 18, 22] the achromatic index of K n was determined for n ≤ 14, when n = p 2 + p + 1 for p an odd prime power and n = q 2 + 2q + 1 − a  ... 
arXiv:1912.10104v1 fatcat:fjazhfxiwrdlrm7emam337xl6e

Achromatic numbers for circulant graphs and digraphs

Gabriela Araujo-Pardo, Juan José Montellano-Ballesteros, Mika Olsen, Christian Rubio
2020 Discussiones Mathematicae Graph Theory  
Finally, we determine the achromatic index for circulant graphs of q 2 + q + 1 vertices when the projective cyclic plane of odd order q exists.  ...  In general, we give the lower bounds α(C 4q 2 +aq+1 (1, a)) ≥ 4q + 1 and dac( − → C 8q 2 +2(a+4)q+a+3 (1, a)) ≥ 4q + 3 when a is a non quadratic residue of Z 4q+1 for graphs and Z 4q+3 for digraphs, and  ...  conference and to publish the proceedings of it in this special issue of DMGT Journal.  ... 
doi:10.7151/dmgt.2327 fatcat:ku3p36vnszgqbbkrxc6ltvnjei

A few results on block-colourings of designs

Sanpei Kageyama, Ying Miao
1996 Hiroshima Mathematical Journal  
A q-block colouring of a design is a partition of its blocks into q colour classes consisting of pairwise disjoint blocks. The chromatic index is the least q for which such a colouring exists.  ...  In this paper the bounds of chromatic indices of designs are given, and some greedy algorithms for colouring the blocks along with the worst-case colourings are discussed.  ...  The research was supported in part by Grant-in-Aid for Scientific Research, The Ministry of Education, Science and Culture under Contract Numbers (A)06302015 and (C)06640311.  ... 
doi:10.32917/hmj/1206127359 fatcat:5f2h5dmpgvdvzb5rivwm3igqhm

Achromatic arboricity on complete graphs [article]

Gabriela Araujo-Pardo, Christian Rubio-Montiel
2021 arXiv   pre-print
In this paper we study the achromatic arboricity of the complete graph. This parameter arises from the arboricity of a graph as the achromatic index arises from the chromatic index.  ...  The achromatic arboricity of a graph G, denoted by A_α(G), is the maximum number of colors that can be used to color the edges of G such that every color class induces a forest but any two color classes  ...  In particular, if q is a prime power there exists a Π q , that arises from finite fields Z q for q prime, and from GF (q), the Galois Field of order q, when q is a prime power.  ... 
arXiv:2103.12225v1 fatcat:d36lxp2pv5dfbmttz6qm5rxf7u

Some Nontrivial Relationships Between A (N+1) & A(N)
english

Ganesh Vishwas Joshi
2014 International Journal of Innovative Research in Science Engineering and Technology  
The trivial relationship known between achromatic indices of K n & K n+1 is A(n+1)≤A(n)+n. In this paper I have obtain some other nontrivial relationship between the numbers A(n+1) , A(n).  ...  The achromatic index of the complete graph K n is denoted by A(n). Before going to the derivation of the relationships one requires the following definitions & results.  ...  The achromatic index ψ ' (G) of a simple graph G is the maximum number of colours used in the edge colouring of G such that the colouring is complete.  ... 
doi:10.15680/ijirset.2014.0311021 fatcat:5lkgnwfkcfhtnaivqqgzr33lze

Page 2 of Mathematical Reviews Vol. , Issue 90C [page]

1990 Mathematical Reviews  
Alan Tucker (1-SUNYS-S) 90c:05082 05C15 Chetwynd, A. G.; Hilton, A. J. W. (4-RDNG); Hoffman, D. G. (1-ABRN-AL) On the A-subgraph of graphs which are critical with respect to the chromatic index. J.  ...  Let x'(G) denote the chromatic index of a graph G; we say that G is critical if G is connected, in Class 2, and x'(G—e) < x'(G) for each edge e.  ... 

Achromatic phase shifters for nulling interferometry

Jean Gay, Yves Rabbia, Etienne Bascou, Georges Otrio
2017 International Conference on Space Optics — ICSO 2000  
These phase shifts must be performed in the thermal infrared domain and must be achromatic so as to enable working on a large bandwidth.  ...  In the search for extra-solar Earth-like planets, nulling interferometry is required.  ...  Disregarding other effects, unequal intensities such as I 1 /I 2 = q, result in a lowered rejection R(q), which is expressed as : q . ) q . 2 q 1 ( 2 ) q 1 ( ) , q ( R        (3.3) For example,  ... 
doi:10.1117/12.2307870 fatcat:ljxre52dujhj7if26bxtyqrw2e

Transforming reflectance spectra into Munsell color space by using prime colors

A. K. Romney, J. T. Fulton
2006 Proceedings of the National Academy of Sciences of the United States of America  
The three simulated prime color sensitivity curves with peaks at 448, 537, and 600 nm and SDs of 30 nm normalized to have equal areas of 100.  ...  Brill, Ti-lien Hsia, Tarow Indow, and Richard A. Young for helpful suggestions on the manuscript. The research was funded in part by National Science Foundation Grant SBR-9531213 (to A.K.R. and W. H.  ...  On the whole, the mean value of 0.9945 for the redundancy index seems a reasonable summary.  ... 
doi:10.1073/pnas.0607488103 pmid:17030816 pmcid:PMC1622884 fatcat:x3va4r6jsfd6rm4f44ajwixeey

Achromatic double phase conjugate mirror

Hongzhi Kong, Mark Cronin-Golomb, Baruch Fischer
1992 Optics Communications  
We show that the double phase conjugate mirror may be made achromatic by using material dispersion to compensate the dispersion of the photorefraetive grating.  ...  It is shown that the bandwidth of the entire visible spectrum may be accommodated in barium titanate crystal. The achromatic conditions found in the experiment are in good agreement with theory.  ...  Acknowledgements This research was supported by Litton Itek Optical Systems, the National Science Foundation through a Presidential Young Investigator Award, Hughes Aircraft Company, and the U.S.  ... 
doi:10.1016/0030-4018(92)90135-e fatcat:27bm4yvydjaclb4sxhnhnt2rku

Graph colourings and partitions

V. Yegnanarayanan
2001 Theoretical Computer Science  
In this paper we have investigated mainly the three colouring parameters of a graph G, viz., the chromatic number, the achromatic number and the pseudoachromatic number.  ...  We have redeÿned the concept of perfect graphs in terms of these parameters and obtained a few results. Some open problems are raised.  ...  sequence c 1 ; : : : ; c n of n colours, the colour to be assigned to v i is the smallest indexed colour not already assigned to one of its lower-indexed neighbouring vertices.  ... 
doi:10.1016/s0304-3975(00)00231-0 fatcat:kcok2lykebbevba2iigpqamjum

Design Of Nonpolarizing Achromatic Beamsplitters With Dielectric Multilayer Coatings

Muamer Zukic, Karl H. Guenther
1989 Optical Engineering: The Journal of SPIE  
S, for example, glass with an index of refraction ns = 1.52, with air A as the ambient medium (nA = 1.00).  ...  T = 50:50% and 30:70% beamsplitters, which are optimized for the 500 to 600 nm wavelength range and angles of incidence of 40 °, 50 °, and 60° using a computer program based on a damped least squares refining  ...  For example, if |a| = V4|Tr(Mp)| > 0.8, then the Chebyshev polynomials for 2 < q < 6 have values |Sq(a)| ^ 1.  ... 
doi:10.1117/12.7976975 fatcat:rlp3dcyb6ngfzgsr7aj5jhzzve

On the Pseudoachromatic Index of the Complete Graph III

M. Gabriela Araujo-Pardo, Juan José Montellano-Ballesteros, Christian Rubio-Montiel, Ricardo Strausz
2018 Graphs and Combinatorics  
On the other hand, if q is even and there exists Π_q we give a complete edge-colouring of K_m_a with (m_a-a)q colours.  ...  Let Π_q be the projective plane of order q , let ψ(m):=ψ(L(K_m)) the pseudoachromatic number of the complete line graph of order m , let a∈{ 3,4,...,q2+1 } and m_a=(q+1)^2-a .  ...  with the achromatic index α(n) := α(L(K n )).  ... 
doi:10.1007/s00373-017-1872-6 fatcat:owvpn7msejdmficmtw4cqzskda

On the edge achromatic numbers of complete graphs

Robert E. Jamison
1989 Discrete Mathematics  
The goals here are more modest: (1) to give a careful study of the best available upper bound on A(n), (2) to add to the constructions which give reasonable lower bounds for A(n), and (3) to contribute  ...  two edges with a common node, one colored cl and the other colored c2.  ...  The achromatic index Let G be a simple graph. A proper (vertex) k-coloring of G is a map of the vertices of G onto a set of k "colors" so that any two adjacent vertices of G receive different colors.  ... 
doi:10.1016/0012-365x(89)90202-1 fatcat:hknfawzjh5emvmwskyn26zz24q

On line colorings of finite projective spaces [article]

Gabriela Araujo-Pardo, György Kiss, Christian Rubio-Montiel, Adrián Vázquez-Ávila
2021 arXiv   pre-print
In this paper, we prove lower and upper bounds on the achromatic and the pseudoachromatic indices of the n-dimensional finite projective space of order q.  ...  The achromatic index α (S) of S is the largest k such that there exists a proper and complete coloring of S with k colors.  ...  On projective spaces It is well-known that, for any n > 2, the n-dimensional finite projective space of order q exists if and only if q is a prime power and it is unique up to isomorphism.  ... 
arXiv:1702.06769v3 fatcat:hwbr2hq6cvg33mfnge4y3wcbpq

Pseudoachromatic and connected-pseudoachromatic indices of the complete graph

M. Gabriela Araujo-Pardo, Christian Rubio-Montiel
2017 Discrete Applied Mathematics  
Related to the pseudoachromatic index we prove that for q a power of 2 and n=q^2+q+1, ψ'(K_n) is at least q^3+2q-3 which improves the bound q^3+q given by Araujo, Montellano and Strausz [J Graph Theory  ...  The pseudoachromatic index of a graph G, denoted by ψ'(G), is the largest k for which the line graph of G has a complete k-coloring.  ...  Acknowledgment The authors wish to thank the anonymous referees of this paper for their helpful remarks and suggestions.  ... 
doi:10.1016/j.dam.2017.03.019 fatcat:kksgm4kulzckrkldfmxmhavpri
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