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## Filters

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

2019
*
arXiv
*
pre-print

Finally, we determine

arXiv:1912.10104v1
fatcat:fjazhfxiwrdlrm7emam337xl6e
*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*...##
###
Achromatic numbers for circulant graphs and digraphs

2020
*
Discussiones Mathematicae Graph Theory
*

Finally, we determine

doi:10.7151/dmgt.2327
fatcat:ku3p36vnszgqbbkrxc6ltvnjei
*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. ...##
###
A few results on block-colourings of designs

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. ...

##
###
Achromatic arboricity on complete graphs
[article]

2021
*
arXiv
*
pre-print

In this paper we study

arXiv:2103.12225v1
fatcat:d36lxp2pv5dfbmttz6qm5rxf7u
*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. ...##
###
Some Nontrivial Relationships Between A (N+1) & A(N)

english

2014
*
International Journal of Innovative Research in Science Engineering and Technology
*

english

*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. ...

##
###
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

2017
*
International Conference on Space Optics — ICSO 2000
*

These phase shifts must be performed in

doi:10.1117/12.2307870
fatcat:ljxre52dujhj7if26bxtyqrw2e
*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, ...##
###
Transforming reflectance spectra into Munsell color space by using prime colors

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. ...

##
###
Achromatic double phase conjugate mirror

1992
*
Optics Communications
*

We show that

doi:10.1016/0030-4018(92)90135-e
fatcat:27bm4yvydjaclb4sxhnhnt2rku
*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. ...##
###
Graph colourings and partitions

2001
*
Theoretical Computer Science
*

In this paper we have investigated mainly

doi:10.1016/s0304-3975(00)00231-0
fatcat:kcok2lykebbevba2iigpqamjum
*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. ...##
###
Design Of Nonpolarizing Achromatic Beamsplitters With Dielectric Multilayer Coatings

1989
*
Optical Engineering: The Journal of SPIE
*

S,

doi:10.1117/12.7976975
fatcat:rlp3dcyb6ngfzgsr7aj5jhzzve
*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. ...##
###
On the Pseudoachromatic Index of the Complete Graph III

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 )). ...

##
###
On the edge achromatic numbers of complete graphs

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. ...

##
###
On line colorings of finite projective spaces
[article]

2021
*
arXiv
*
pre-print

In this paper, we prove lower and upper bounds

arXiv:1702.06769v3
fatcat:hwbr2hq6cvg33mfnge4y3wcbpq
*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. ...##
###
Pseudoachromatic and connected-pseudoachromatic indices of the complete graph

2017
*
Discrete Applied Mathematics
*

Related to

doi:10.1016/j.dam.2017.03.019
fatcat:kksgm4kulzckrkldfmxmhavpri
*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. ...
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