• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.313-323
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• 2007

The equitable total chromatic number ${\chi}_{et}(G)$ of a graph G is the smallest integer ${\kappa}$ for which G has a total ${\kappa}$-coloring such that the number of vertices and edges in any two color classes differ by at most one. In this paper, we determine the equitable total chromatic number of one class of the graphs.

• KSTLE International Journal
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• v.8 no.1
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• pp.1-6
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• 2007

Chromatic device was developed using light emitting diodes, optic fibers and photodiodes. Chromatic ratio and total contamination parameters based on transmitted light intensity in Red, Green, and Blue wavelengths were used for oil chemical and particulate contamination assessment. Chromatic ratio criterion was found independent of the particulate contamination of oil; but depended on chemical degradation, being more sensitive for synthetic than mineral hydraulic oil. Total contamination index of the sensor depended on both the chemical degradation and particulate contamination of the oil; being most sensitive in blue wavelength, and least in the red. Test results for synthetic hydraulic oils monitored corroborated with results of other tests such as viscosity, total acid number, elemental optical emission spectroscopy, particulate counts and UV-VIS photospectrometry. Chromatic ratio showed a clearer indication of oil degradation, compared to key monitoring parameters such as total acid number, viscosity and particle counts. The results showed that these parameters are effective criteria for the condition monitoring of synthetic hydraulic oils.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.567-582
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• 2024

A total coloring of a graph G is an assignment of colors to the elements of a graphs G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G , denoted by χ''(G), is the minimum number of colors that suffice in a total coloring. In this paper, we proved the Behzad and Vizing conjecture for certain convex polytope graphs D^p_n, Q^p_n, R^p_n, E_n, S_n, G_n, T_n, U_n, C_n,respectively. This significant result in a graph G contributes to the advancement of graph theory and combinatorics by further confirming the conjecture's applicability to specific classes of graphs. The presented proof of the Behzad and Vizing conjecture for certain convex polytope graphs not only provides theoretical insights into the structural properties of graphs but also has practical implications. Overall, this paper contributes to the advancement of graph theory and combinatorics by confirming the validity of the Behzad and Vizing conjecture in a graph G and establishing its relevance to applied problems in sciences and engineering.

• 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.53 no.1
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• pp.139-151
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• 2016

A k-total-coloring of a graph G is a coloring of $V{\cup}E$ using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number ${\chi}^{{\prime}{\prime}}(G)$ of G is the smallest integer k such that G has a k-total-coloring. Let G be a planar graph with maximum degree ${\Delta}$. In this paper, it's proved that if ${\Delta}{\geq}7$ and G does not contain adjacent 5-cycles, then the total chromatic number ${\chi}^{{\prime}{\prime}}(G)$ is ${\Delta}+1$.

• Journal of applied mathematics & informatics
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• v.42 no.2
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• pp.353-366
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• 2024

A total coloring of a graph G is an assignment of colors to both the vertices and edges of G, such that no two adjacent or incident vertices and edges of G are assigned the same colors. In this paper, we have discussed the total coloring of M(T_n), M(D_n), M(DT_n), M(AT_n), M(DA(T_n)), M(Q_n), M(AQ_n) and also obtained the total chromatic number of M(T_n), M(D_n), M(DT_n), M(AT_n), M(DA(T_n)), M(Q_n), M(AQ_n).

• 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 the Korean Society of Tobacco Science
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• v.26 no.1
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• pp.27-34
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• 2004

This study was carried out to examinate the effect of aging period on chromatic, chemical and organoleptic characteristics, and to evaluate of optimum aging period for each grade in flue-cured leaf tobacco. The leaf tobaccos were produced in 2000, and threshed, redried and packed in carton box under the current methods. Four grades of processed leaf(A3O, B1O, C1L and D3L) were stored during 24 months(May 10, 2001 to April 31, 2003) in warehouse of Chungju Leaf Tobacco Processing Factory. The leaf tobaccos were sampled at three month intervals for analysis of chromatic, chemical and organoleptic properties. Yellow(b), pH values and total sugar contents of four grades were significantly decreased during the aging. Filling values, tar, nicotine and CO contents of tobacco smoke, and puff number of cigarettes were not significantly changed during the aging. Positive correlation coefficients were significantly observed between taste and irritation of the calculated attributes from contents of volatile oil components in leaf tobacco and those of the panel sensory attributes. The ratio of maximum change in taste attribute was larger than that in irritation attribute during aging. The optimum aging periods estimated by taste for A3O, B1O, C1L and D3L were 17.8, 14.9, 10.8, and 9.8 months, respectively. The thin leaf(Primings and Cutters) undergo satisfactory aging earlier as compared to bodied leaf(Leaf and Tips). The results suggest that decrease of aging period for thin leaf from 18～21 to 9～12 months may be beneficial to save storage cost in flue-cured tobacco.

• Proceedings of the Safety Management and Science Conference
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• 2003.05a
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• pp.135-141
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• 2003

An experiment was conducted to identify salient colours for information coding at electronic video displays. CRT and TFT-LCD were used to evaluate the effect of different types of electronic displays on the salience of colours. Total of 100 Subjects, 50 for each display were asked to select more salient 10 colours among 24 given colours. There was no statistically significant difference in the salience of colours between the two display types. The result showed that the tested colours could be clustered into 5 categories according to their brightness, saturation, and the number of R, G, B elements occupied to reproduce the colours. Three achromatic colours (black, white, grey) and eight chromatic colors (red, yellow, green, blue, cyan, orange, magenta, and indigo) were identified as the salient colours at the electronic video displays. The result also showed that the eight chromatic colours could be clustered into two distinct categories, landmark colours(red, yellow, green, and blue) and the other basic colours (cyan, magenta, orange, and indigo). It is noticeable that cyan, magenta, and indigo substituted for pink, purple, and brown that were recommended as the salient colours for the environment not using electronic video displays by the previous researches.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.14 no.5
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• pp.263-269
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• 2014

In this paper, I disprove Hadwiger conjecture of the vertex coloring problem, which asserts that "All $K_k$-minor free graphs can be colored with k-1 number of colors, i.e., ${\chi}(G)=k$ given $K_k$-minor." Pursuant to Hadwiger conjecture, one shall obtain an NP-complete k-minor to determine ${\chi}(G)=k$, and solve another NP-complete vertex coloring problem as a means to color vertices. In order to disprove Hadwiger conjecture in this paper, I propose an algorithm of linear time complexity O(V) that yields the exact solution to the vertex coloring problem. The proposed algorithm assigns vertex with the minimum degree to the Maximum Independent Set (MIS) and repeats this process on a simplified graph derived by deleting adjacent edges to the MIS vertex so as to finally obtain an MIS with a single color. Next, it repeats the process on a simplified graph derived by deleting edges of the MIS vertex to obtain an MIS whose number of vertex color corresponds to ${\chi}(G)=k$. Also presented in this paper using the proposed algorithm is an additional algorithm that searches solution of ${\chi}^{{\prime}{\prime}}(G)$, the total chromatic number, which also remains NP-complete. When applied to a $K_4$-minor graph, the proposed algorithm has obtained ${\chi}(G)=3$ instead of ${\chi}(G)=4$, proving that the Hadwiger conjecture is not universally applicable to all the graphs. The proposed algorithm, however, is a simple algorithm that directly obtains an independent set minor of ${\chi}(G)=k$ to assign an equal color to the vertices of each independent set without having to determine minors in the first place.