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NEIGHBORHOOD CONDITION AND FRACTIONAL f-FACTORS IN GRAPHS
Liu, Hongxia,Liu, Guizhen The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5
Let G be a graph with vertex set V(G) and let f be a nonnegative integer-valued function defined on V(G). A spanning subgraph F of G is called a fractional f-factor if $d^h_G$(x)=f(x) for all x $\in$ for all x $\in$ V (G), where $d^h_G$ (x) = ${\Sigma}_{e{\in}E_x}$ h(e) is the fractional degree of x $\in$ V(F) with $E_x$ = {e : e = xy $\in$ E|G|}. In this paper it is proved that if ${\delta}(G){\geq}{\frac{b^2(k-1)}{a}},\;n>\frac{(a+b)(k(a+b)-2)}{a}$ and $|N_G(x_1){\cup}N_G(x_2){\cup}{\cdots}{\cup}N_G(x_k)|{\geq}\frac{bn}{a+b}$ for any independent subset ${x_1,x_2,...,x_k}$ of V(G), then G has a fractional f-factor. Where k $\geq$ 2 be a positive integer not larger than the independence number of G, a and b are integers such that 1 $\leq$ a $\leq$ f(x) $\leq$ b for every x $\in$ V(G). Furthermore, we show that the result is best possible in some sense.
NEIGHBORHOOD CONDITION AND FRACTIONAL f-FACTORS IN GRAPHS
Hongxia Liu,Guizhen Liu 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
Let G be a graph with vertex set V (G) and let f be a nonnegative integer-valued function defined on V (G). A spanning subgraph F of G is called a fractional f-factor if dhG(x) = f(x) for all x ∈ V (G), where dhG(x) = ∑e2Ex h(e) is the fractional degree of x ∈ V (F) with Ex = {e : e = xy ∈ E(G)}. In this paper it is proved that if δ(G) ≥ <수식>, n > <수식> and |NG(x1)∪ NG(x2) ∪ ·· · ∪ NG(xk)|≥ <수식> for any independent subset {x1, x2, ..., xk} of V (G), then G has a fractional f-factor . Where k≥ 2 be a positive integer not larger than the independence number of G, a and b are integers such that 1≤ a≤ f(x) ≤b for every x ∈ V (G). Furthermore, we show that the result is best possible in some sense. Let G be a graph with vertex set V (G) and let f be a nonnegative integer-valued function defined on V (G). A spanning subgraph F of G is called a fractional f-factor if dhG(x) = f(x) for all x ∈ V (G), where dhG(x) = ∑e2Ex h(e) is the fractional degree of x ∈ V (F) with Ex = {e : e = xy ∈ E(G)}. In this paper it is proved that if δ(G) ≥ <수식>, n > <수식> and |NG(x1)∪ NG(x2) ∪ ·· · ∪ NG(xk)|≥ <수식> for any independent subset {x1, x2, ..., xk} of V (G), then G has a fractional f-factor . Where k≥ 2 be a positive integer not larger than the independence number of G, a and b are integers such that 1≤ a≤ f(x) ≤b for every x ∈ V (G). Furthermore, we show that the result is best possible in some sense.
Tannic Acid and Ferrous Sulfate Modified Kapok Fiber for Oil-water Separation
Guizhen Ke,Shuhui Chen,Jiani Tang,Huanmin Li,Keshuai Liu 한국섬유공학회 2022 Fibers and polymers Vol.23 No.11
Biodegradable oil absorbing materials have attracted extensive attention in dealing with oil spill pollution. Todevelop more excellent oleophilic oil sorbent, the combination of tannic acid and ferrous sulfate was employed to treat therenewable and biodegradable natural kapok fiber. The as-prepared kapok showed superhydrophobic characteristics with highwater contact for the formation of tannic acid and ferrous ion complex on kapok surface, which was confirmed by scanningelectron microscopy observation and infrared spectroscopy and X-ray diffraction analysis. The preparation process of themodified kapok fiber was optimized. When the sodium chlorite pretreated kapok fiber was firstly treated with 4.25 g/l tannicacid and subsequently immersed in 4.25 g/l tannic acid and 6.25 g/l ferrous sulfate mixture at 60 °C for 15 min, the aspreparedkapok fiber presented the best oil absorption capacity and hydrophobicity (water contact angle 150.7 °). The treatedkapok had good adsorption capacity for oils and organic solvents, 48.2, 41.6, 53.4, 52.8, 49.2, 41.2 g/g for vegetable oil,diesel oil, silicone oil, carbon tetrachloride, DMF and n-hexane, respectively. In addition, the tannic acid and ferrous sulfatetreated kapok was recycled for six times through simple mechanical squeezing, and showed good oil-water separationperformance oil, indicating its potential application in the removal of spilled oil on water.
A Comparative Study of the Social Security System between China and Korea
Guizhen Heng,Jing Li Liu,장세철 한국비교정부학회 2009 한국비교정부학보 Vol.13 No.2
With the continuous development of economy and the improvement of social progress, most countries in the world establish a more comprehensive social security system and continuously improve its functions. Improving social security system is one of the most important problems that China as a socialist country has to solve urgently at present and it is also a guarantee of building a harmonious socialist society. The social security system in the newly industrialized country, South Korea, especially deserves our attention. We should also see that due to the difference of the productivity level of development, financial and economic strength, population and resources the social security system behave divercitily some of the institutional framework and policy system of referential significance to us. This article thinks that if China can learn such experiences from South Korea as legislature first, avoiding a financial crisis, the focus on protection, making full use of the family security features, etc., then they can play a positive role in promoting the improvement of China's social security system.
LIST EDGE AND LIST TOTAL COLORINGS OF PLANAR GRAPHS WITHOUT 6-CYCLES WITH CHORD
Aijun Dong,Guizhen Liu,Guojun Li 대한수학회 2012 대한수학회보 Vol.49 No.2
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 ${\Delta}(G){\geq}6$.
SOME RESULTS ON BINDING NUMBER AND FRACTIONAL PERFECT MATCHING
Zhu, Yan,Liu, Guizhen 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.25 No.1
The relationships between binding number and fractional edge (vertex)-deletability or fractional k-extendability of graphs are studied. Furthermore, we show that the result about fractional vertex-deletability are best possible.
(1, λ)-EMBEDDED GRAPHS AND THE ACYCLIC EDGE CHOOSABILITY
Xin Zhang,Guizhen Liu,Jian-Liang Wu 대한수학회 2012 대한수학회보 Vol.49 No.3
A (1, λ)-embedded graph is a graph that can be embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge. A graph $G$ is called α-linear if there exists an integral constant β such that $e(G^{\prime}){\leq}{\alpha}v(G^{\prime})+{\beta}$ for each $G^{\prime}{\subseteq}G$. In this paper, it is shown that every (1, λ)-embedded graph $G$ is 4-linear for all possible λ, and is acyclicly edge-($3{\Delta}(G)+70$)-choosable for λ = 1, 2.