| On All Fractional （a,b, k）-Critical Graphs |
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| 作者姓名： | Si Zhong ZHOU Zhi Ren SUN |
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| 作者单位： | [1]School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 212003, P. R. China [2]School of Mathematical Sciences, Nanjing Normal University, Nanjing 210048, P. R. China |
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| 基金项目： | Supported by National Natural Science Foundation of China(Grant No.11371009) |
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| 摘 要： | Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.
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| 关 键 词： | 临界图 分数 非负整数 使用结果 子集 下界 |
On all fractional (a,b, k)-critical graphs |
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| Si Zhong ZHOU,Zhi Ren SUN.On all fractional (a,b, k)-critical graphs[J].Acta Mathematica Sinica,2014,30(4):696-702. |
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| Authors: | Si Zhong Zhou Zhi Ren Sun |
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| Institution: | 1. School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, 212003, P. R. China
2. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210046, P. R. China
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| Abstract: | Let a, b, k, r be nonnegative integers with 1 ≤ a ≤ b and r ≥ 2. Let G be a graph of order n with $n > \tfrac{{(a + b)(r(a + b) - 2) + ak}} {a} $ . In this paper, we first show a characterization for all fractional (a, b, k)-critical graphs. Then using the result, we prove that G is all fractional (a, b, k)-critical if $\delta (G) \geqslant \tfrac{{(r - 1)b^2 }} {a} + k $ and $|N_G (x_1 ) \cup N_G (x_2 ) \cup \cdots \cup N_G (x_r )| \geqslant \tfrac{{bn + ak}} {{a + b}} $ for any independent subset {x 1, x 2, …, x r } in G. Furthermore, it is shown that the lower bound on the condition $|N_G (x_1 ) \cup N_G (x_2 ) \cup \cdots \cup N_G (x_r )| \geqslant \tfrac{{bn + ak}} {{a + b}} $ is best possible in some sense, and it is an extension of Lu’s previous result. |
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| Keywords: | Graph neighborhood union all fractional [a b]-factor all fractional(a b k)-critical |
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