• Honam Mathematical Journal
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• v.27 no.4
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• pp.551-559
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• 2005

A permutation graph over a graph G is a generalization of both a graph bundle and a graph covering over G. In this paper, we characterize the F-permutation graphs over a graph whose chromatic numbers are 2. We determine the chromatic numbers of $C_n$-permutation graphs over a tree and the $K_m$-permutation graphs over a cycle.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.105-115
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• 2011

Let f be a function which assigns a positive integer f(v) to each vertex v $\in$ V (G), let r, s and t be non-negative integers. An f-coloring of G is an edge-coloring of G such that each vertex v $\in$ V (G) has at most f(v) incident edges colored with the same color. The minimum number of colors needed to f-color G is called the f-chromatic index of G and denoted by ${\chi}'_f$(G). An [r, s, t; f]-coloring of a graph G is a mapping c from V(G) $\bigcup$ E(G) to the color set C = {0, 1, $\ldots$; k - 1} such that |c($v_i$) - c($v_j$ )| $\geq$ r for every two adjacent vertices $v_i$ and $v_j$, |c($e_i$ - c($e_j$)| $\geq$ s and ${\alpha}(v_i)$ $\leq$ f($v_i$) for all $v_i$ $\in$ V (G), ${\alpha}$ $\in$ C where ${\alpha}(v_i)$ denotes the number of ${\alpha}$-edges incident with the vertex $v_i$ and $e_i$, $e_j$ are edges which are incident with $v_i$ but colored with different colors, |c($e_i$)-c($v_j$)| $\geq$ t for all pairs of incident vertices and edges. The minimum k such that G has an [r, s, t; f]-coloring with k colors is defined as the [r, s, t; f]-chromatic number and denoted by ${\chi}_{r,s,t;f}$ (G). In this paper, we present some general bounds for [r, s, t; f]-coloring firstly. After that, we obtain some important properties under the restriction min{r, s, t} = 0 or min{r, s, t} = 1. Finally, we present some problems for further research.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.127-133
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• 2005

An f-coloring of a graph G = (V, E) is a coloring of edge set E such that each color appears at each vertex v $\in$ V at most f(v) times. The minimum number of colors needed to f-color G is called the f-chromatic index $\chi'_f(G)$ of G. Any graph G has f-chromatic index equal to ${\Delta}_f(G)\;or\;{\Delta}_f(G)+1,\;where\;{\Delta}_f(G)\;=\;max\{{\lceil}\frac{d(v)}{f(v)}{\rceil}\}$. If $\chi'_f(G)$= ${\Delta}$f(G), then G is of $C_f$ 1 ; otherwise G is of $C_f$ 2. In this paper, the classification problem of complete graphs on f-coloring is solved completely.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.357-366
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• 2007

Let G(V, E) be a simple graph, and let f be an integer function on V with $1{\leq}f(v){\leq}d(v)$ to each vertex $v{\in}V$. An f-edge cover-coloring of a graph G is a coloring of edge set E such that each color appears at each vertex $v{\in}V$ at least f(v) times. The f-edge cover chromatic index of G, denoted by ${\chi}'_{fc}(G)$, is the maximum number of colors such that an f-edge cover-coloring of G exists. Any simple graph G has an f-edge cover chromatic index equal to ${\delta}_f\;or\;{\delta}_f-1,\;where\;{\delta}_f{=}^{min}_{v{\in}V}\{\lfloor\frac{d(v)}{f(v)}\rfloor\}$. Let G be a connected and not complete graph with ${\chi}'_{fc}(G)={\delta}_f-1$, if for each $u,\;v{\in}V\;and\;e=uv{\nin}E$, we have ${\chi}'_{fc}(G+e)>{\chi}'_{fc}(G)$, then G is called an f-edge covered critical graph. In this paper, some properties on f-edge covered critical graph are discussed. It is proved that if G is an f-edge covered critical graph, then for each $u,\;v{\in}V\;and\;e=uv{\nin}E$ there exists $w{\in}\{u,v\}\;with\;d(w)\leq{\delta}_f(f(w)+1)-2$ such that w is adjacent to at least $d(w)-{\delta}_f+1$ vertices which are all ${\delta}_f-vertex$ in G.

• Korean Journal of Optics and Photonics
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• v.11 no.4
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• pp.232-238
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• 2000

The new collimation testing technique of a white light beam using vernier Moire fringes of two line or circular gratings with different pitches is presented. We can visually measure the defocusing ($\Deltaf$), the divergence angle ($\theta$), and the longitudinal chromatic aberration $(L_{ch})$ of a collimating lens by using the technique. For example, we obtained $\Deltaf$= 21.9 mm and $\theta=0.0038^{\circ}$ for a testing lens with the focallengthf = 120.0 mm and F-number of F/2.4. The longitudinal chromatic aberration $L_{ch}$ of another testing lens withf = 65.0 mm, F/1.6, and the Abbe number V = 64.1 for the incident wavelengths of $\lambda_1=480 nm and \lambda_2=640 nm$ is easily measured by same technique. It is found that the measured value $L_{ch}=1.59mm(\pm0.01mm)$ is well agreed with $L_{ch}=1.58mm(\pm0.01mm)$ obtained by the autofocus method.

• Journal of the Optical Society of Korea
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• v.1 no.2
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• pp.74-80
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• 1997

300mm F/4.0 telephotolens with diffractive hybrid lens was designed, and its optical performance was tested and compared with a traditional lens system. DOE(Diffractive Optical Element) reconstructs wavefronts using wave phenomena of light to focus the incident light onto the focal point and has negative Abbe number while a traditional lens uses geometrical phenomena of light and has positive Abbe number. Therefore, a diffractive hybrid lens containing both refractive and diffractive elements can remarkably correct chromatic aberration and spherical aberration of an optical system. We investigated and analyzed the optical properties of a diffractive hybrid lens for the visible spectrum, and we used a difractive hybrid lens to design and evaluate a 300mm F/4.0 telephotolens without the special LD(Low Dispersive) glass lens which is costly and difficult to manufacture. Most traditional telephotolenses use the special LD glass for chromatic aberration correcton. Optical performance tests such as resolution and characteristics of aberration of both lens systems using a diffractive hybrid lens and traditional lens were performed.

• Korean Journal of Optics and Photonics
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• v.31 no.6
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• pp.281-289
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• 2020

This paper reports a slim mobile lens design using a hybrid refractive/diffractive optical element. Conventionally a wide field of view (FOV) camera-lens design adopts a retrofocus type having a negative (-) lens at the forefront, so that it improves in imaging performance over the wide FOV, but with the sacrifice of longer total track length (TTL). However, we chose a telephoto type as a baseline design layout having a positive (+) lens at the forefront, to achieving slimness, based on the specification analysis of 23 reported optical designs. Following preliminary optimization of a baseline design and aberration analysis based on Zernike-polynomial decomposition, we applied a hybrid refractive/diffractive element to effectively reduce the residual chromatic spherical aberration. The optimized optical design consists of 6 optical elements, including one hybrid element. It results in a very slim telephoto ratio of 1.7, having an f-number of 2.0, FOV of 90°, effective focal length of 2.23 mm, and TTL of 3.7 mm. Compared to a comparable conventional lens design with no hybrid elements, the hybrid design improved the value of the modulation transfer function (MTF) at a spatial frequency of 180 cycles/mm from 63% to 71-73% at zero field (0 F), and about 2-3% at 0.5, 0.7, and 0.9 fields. It was also found that a design with a hybrid lens with only two diffraction zones at the stop achieved the same performance improvement.

• Korean Journal of Optics and Photonics
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• v.11 no.4
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• pp.239-245
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• 2000

In this paper, the fundamental properties of diffractive optical element were investigated. Also, this work deals with theoretical approaches for achromatization in DOE's optical system based on thin lens theory. It is found that achromatization could be satisfied by one hybrid lens only, which is composed of a diffractive and a refractive element. In order to have compact optical system, we used the tele-photo type lens composed of a positive and a negative power elements instead of retro-focus lens. From the Gaussian brackets and Seidel aberration theory, the initial design was numerically obtained. The aberration properties of an initial design was aplanat and flat field. In order to correct the chromatic aberrations, refractive and diffractive elements were used on front element. This hybrid lens is also useful for correction of higher order aberrations. Compared to conventional design composed of refractive lenses only, this approach dramatically improved the compactness of the optical system. Finally, residual aberration balancing results in a lens with focal length of 3.89 mm and overall length of 5.19 mm, which has enough performance over an f-number of 4.0. Also, it is expected to fulfill all the requirements of a digital still camera lens. This optical system is superior to the current refractive lens system in the number of elements, weight, and aberration properties. rties.

• Korean Journal of Optics and Photonics
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• v.9 no.3
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• pp.134-141
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• 1998

A high performance afocal zoom telescope has been designed to operate in the 7.6${\mu}{\textrm}{m}$ to 10.3${\mu}{\textrm}{m}$ waveband for thermal imaging system. This IR zoom telescope is characterized by using of two movable optical element groups, variator and compensator, with mechanically compensated method and the positioning of these groups is controlled by means of a computerized program. The optical performance over the entire 4:1 zoom range and $\pm$2.31~$\pm$9.36 degrees field of view is near diffraction limit while maintaining a constant F-number. The all refracting surfaces of this system except only one aspheric surface are spherical curvature and the material for the optical elements is selected Ge and ZnSe which is used for correction of chromatic aberration.