• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.263-271
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• 2024

A vertex coloring of a graph G is called injective if any two vertices with a common neighbor receive distinct colors. A graph G is injectively k-choosable if any list L of admissible colors on V (G) of size k allows an injective coloring 𝜑 such that 𝜑(v) ∈ L(v) whenever v ∈ V (G). The least k for which G is injectively k-choosable is denoted by χ^l_i(G). For a planar graph G, Bu et al. proved that χ^l_i(G) ≤ ∆ + 6 if girth g ≥ 5 and maximum degree ∆(G) ≥ 8. In this paper, we improve this result by showing that χ^l_i(G) ≤ ∆ + 6 for g ≥ 5 and arbitrary ∆(G).

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.415-430
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• 2018

A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment $L=\{L(v):v{\in}V(G)\}$, there exists an acyclic coloring ${\phi}$ of G such that ${\phi}(v){\in}L(v)$ for all $v{\in}V(G)$ A graph G is acyclically k-choosable if G is acyclically L-list colorable for any list assignment with $L(v){\geq}k$ for all $v{\in}V(G)$. Let G be a planar graph without 5-cycles and adjacent 4-cycles. In this article, we prove that G is acyclically 5-choosable if every vertex v in G is incident with at most one i-cycle, $i {\in}\{6,7\}$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.359-365
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• 2012

Giving a planar graph G, let $x^'_l(G)$ and $x^{''}_l(G)$ denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if a planar graph G without 6-cycles with chord, then $x^'_l(G){\leq}{\Delta}(G)+1$ and $x^{''}_l(G){\leq}{\Delta}(G)+2$ where ${\Delta}(G){\geq}6$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1303-1314
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• 2013

An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors. A graph G is said to be injectively $k$-choosable if any list $L(v)$ of size at least $k$ for every vertex $v$ allows an injective coloring ${\phi}(v)$ such that ${\phi}(v){\in}L(v)$ for every $v{\in}V(G)$. The least $k$ for which G is injectively $k$-choosable is the injective choosability number of G, denoted by ${\chi}^l_i(G)$. In this paper, we obtain new sufficient conditions to be ${\chi}^l_i(G)={\Delta}(G)$. Maximum average degree, mad(G), is defined by mad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that if mad(G) < $\frac{8k-3}{3k}$, then ${\chi}^l_i(G)={\Delta}(G)$ where $k={\Delta}(G)$ and ${\Delta}(G){\geq}6$. In addition, when ${\Delta}(G)=5$ we prove that ${\chi}^l_i(G)={\Delta}(G)$ if mad(G) < $\frac{17}{7}$, and when ${\Delta}(G)=4$ we prove that ${\chi}^l_i(G)={\Delta}(G)$ if mad(G) < $\frac{7}{3}$. These results generalize some of previous results in [1, 4].

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.149-156
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• 2006

The circular list coloring is a circular version of list colorings of graphs. Let $\chi_{c,l}$ denote the circular choosability(or the circular list chromatic number). In this paper, the circular choosability of outer planar graphs and odd wheel is discussed.

• Korean Institute of Interior Design Journal
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• no.14
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• pp.19-26
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• 1998

The purpose of this research is to examine the scope and effectiveness of the utilization of computer programs in universities and educational institutions teaching and training subjects related to interior construction such as interior and exterior design display coloring furniture design perspective drawing expression technique space planning lighting design and portfolio production. The list of currently available non-CAD computer programs for these purpose and the effectiveness of utilizing adobe photoshop in interior design will be examined.

• Journal of Korean Society on Water Environment
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• v.21 no.5
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• pp.484-489
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• 2005

In order to determine the removal rate of non-biodegradable substances and the change of their structural properties, this study was carried out by an ozone-treatment experiment on leachate collected from the landfill area of D City in Chung chung nam-do and examined the change of the chemical properties of non-biodegradable substances. The main elements of non-biodegradable substances in landfill leachate were benzene, toluene, trichloroethane, trichloroethylene, xylene, etc. and the concentration of toluene was 15.7 mg/L on the average, benzene 7.2 mg/L, trichloroethane 1.1 mg/L, trichloroethylene 0.75 mg/L and xylene 0.5 mg/L. When leachate was treated with ozone for 10 min, 30 min and 60 min, UV absorbance was reduced with the increase of reaction time, and the reduction rate was 38.6% at 60 min. TOC was removed by 13.2% at 60 min. The low reduction rate of TOC may be because TOC reacts indirectly with OH radical produced from reaction with ozone while UV absorbance usually relies on direct reaction between organic matters and ozone molecules. Color was removed by up to 97%, which suggests that ozonation is highly effective in removing coloring elements in leachate. Sixteen kinds of non-biodegradable compounds were found in the leachate and most of them had the characteristic of aromatic hydrocarbon. Among them dibutyl phthalate was identical with a substance included in the list of US EPA, which is classified as a mutagen that may cause the mutation of genes and disorders in chromosomes. In addition, 2,5-Cyclohexadiene-1,4-dione, 1,2-Benzenedicarboxylic acid and butyl octyl ester were found to be similar to substances listed by USEPA. According to the result of analyzing structural changes before and after ozonation using GC-MS, cyclic compounds and aromatic compounds were observed in the original water and aliphatic compounds were newly observed after ozonation. In addition, through ozonation, humic substances of high molecular weight were oxidized and decomposed and produced low-molecular compounds such as aldehyde, ketone and carboxyl acid and highly biodegradable aliphatic carbon, which suggests the bio-degradability of non-biodegradable substances.