A Degree Condition for Graphs Having All (a, b)-parity Factors

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED Acta Mathematicae Applicatae Sinica, English Series Pub Date : 2024-06-05 DOI:10.1007/s10255-024-1090-y
Hao-dong Liu, Hong-liang Lu
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Abstract

Let a and b be positive integers such that ab and ab (mod 2). We say that G has all (a, b)-parity factors if G has an h-factor for every function h: V(G) → {a, a + 2, ⋯, b − 2, b} with bV(G)∣ even and h(v) ≡ b (mod 2) for all vV(G). In this paper, we prove that every graph G with n ≥ 2(b + 1)(a + b) vertices has all (a, b)-parity factors if δ(G) ≥ (b2b)/a, and for any two nonadjacent vertices \(u,\,v\, \in \,V\,(G),\,\max \{{d_G}(u),\,{d_G}(v)\} \, \ge {{bn} \over {a + b}}\). Moreover, we show that this result is best possible in some sense.

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具有所有 (a, b) 奇偶因子的图的度条件
设 a 和 b 为正整数,且 a≤b 和 a≡b (mod 2)。如果对于每个函数 h,G 都有一个 h 因子,那么我们就说 G 具有所有 (a, b) 奇偶因子:V(G)→{a,a + 2,⋯,b - 2,b},其中 b∣V(G)∣ 偶数,且对于所有 v∈V(G) ,h(v) ≡ b(mod 2)。在本文中,我们将证明,如果 δ(G) ≥ (b2 - b)/a, 并且对于任意两个非相邻顶点 \(u,\,v\, \in \,V\,(G),\,\max \{{d_G}(u),\,{d_G}(v)\} ,则具有 n≥ 2(b + 1)(a + b) 个顶点的每个图 G 都具有所有(a, b)奇偶因子。\ge {{bn}\over {a + b}})。此外,我们还证明了这一结果在某种意义上是最好的。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
70
审稿时长
3.0 months
期刊介绍: Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.
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