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学者姓名：周垂香
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Abstract ：
The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a matching. It has been proved independently by different groups of people that every connected cubic graph can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges. In this paper, we establish a bound on the number of paths of two edges, proving that every connected cubic graph on n vertices can be decomposed into a spanning tree, a set of cycles, and a set of vertex-disjoint paths of at most two edges such that the number of paths of two edges is at most n-4 6 . Our proof is based on a structural analysis, which might provide a new approach to attack the 3-Decomposition Conjecture. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Keyword ：
3-Decomposition Conjecture 3-Decomposition Conjecture Cubic graph Cubic graph Graph decomposition Graph decomposition
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| GB/T 7714 | Fan, Genghua , Zhou, Chuixiang . Hoffmann-Ostenhof's 3-Decomposition Conjecture [J]. | DISCRETE MATHEMATICS , 2025 , 348 (7) . |
| MLA | Fan, Genghua 等. "Hoffmann-Ostenhof's 3-Decomposition Conjecture" . | DISCRETE MATHEMATICS 348 . 7 (2025) . |
| APA | Fan, Genghua , Zhou, Chuixiang . Hoffmann-Ostenhof's 3-Decomposition Conjecture . | DISCRETE MATHEMATICS , 2025 , 348 (7) . |
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Abstract ：
Let G be a 10-regular graph which does not contain any 4-cycles. In this paper, we prove that G can be decomposed into paths of length 5, such that every vertex is a terminal of exactly two paths.
Keyword ：
10-regular graph 10-regular graph decomposition decomposition path path
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| GB/T 7714 | Xie, Mengmeng , Zhou, Chuixiang . Decomposing 10-Regular Graphs into Paths of Length 5 [J]. | DISCUSSIONES MATHEMATICAE GRAPH THEORY , 2022 , 42 (4) : 1089-1097 . |
| MLA | Xie, Mengmeng 等. "Decomposing 10-Regular Graphs into Paths of Length 5" . | DISCUSSIONES MATHEMATICAE GRAPH THEORY 42 . 4 (2022) : 1089-1097 . |
| APA | Xie, Mengmeng , Zhou, Chuixiang . Decomposing 10-Regular Graphs into Paths of Length 5 . | DISCUSSIONES MATHEMATICAE GRAPH THEORY , 2022 , 42 (4) , 1089-1097 . |
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Let G be a 2k-regular graph in which every cycle of length at most k is an induced cycle. In this paper, we prove that G can be decomposed into paths of length k, and moreover, every vertex is a terminal of exactly two paths.
Keyword ：
2k-regular graph 2k-regular graph Decomposition Decomposition Path Path Trail Trail
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| GB/T 7714 | Xie, Mengmeng , Zhou, Chuixiang . Decomposing 2k-Regular Graphs into Paths of Length k [J]. | GRAPHS AND COMBINATORICS , 2022 , 38 (5) . |
| MLA | Xie, Mengmeng 等. "Decomposing 2k-Regular Graphs into Paths of Length k" . | GRAPHS AND COMBINATORICS 38 . 5 (2022) . |
| APA | Xie, Mengmeng , Zhou, Chuixiang . Decomposing 2k-Regular Graphs into Paths of Length k . | GRAPHS AND COMBINATORICS , 2022 , 38 (5) . |
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Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let p(G) denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then p(G) <= inverted right perpendicularn/2inverted left perpendicular. If G is allowed to be disconnected, then the upper bound left perpendicular3/4nright perpendicular for p(G) was obtained by Donald [7], which was improved to left perpendicular2/3nright perpendicular independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, left perpendicular2/3nright perpendicular is reached and so this bound is tight. If triangles are forbidden in G, then p(G) <= left perpendicularg+1/2g nright perpendicular can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that p(G) <= left perpendicular3n/5right perpendicular, which improves the above result with g = 4. (C) 2022 Elsevier B.V. All rights reserved.
Keyword ：
Decomposition Decomposition Gallai & rsquo;s conjecture Gallai & rsquo;s conjecture Path Path Triangle-free Triangle-free
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| GB/T 7714 | Chu, Yanan , Fan, Genghua , Zhou, Chuixiang . Path decompositions of triangle-free graphs [J]. | DISCRETE MATHEMATICS , 2022 , 345 (7) . |
| MLA | Chu, Yanan 等. "Path decompositions of triangle-free graphs" . | DISCRETE MATHEMATICS 345 . 7 (2022) . |
| APA | Chu, Yanan , Fan, Genghua , Zhou, Chuixiang . Path decompositions of triangle-free graphs . | DISCRETE MATHEMATICS , 2022 , 345 (7) . |
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Abstract ：
Let (G, sigma) be a 2-edge-connected flow-admissible signed graph. In this paper, we prove that (G, sigma) has a signed circuit cover with length at most 3 vertical bar E(G)vertical bar.
Keyword ：
Flow-admissible Flow-admissible Signed circuit cover Signed circuit cover Signed graph Signed graph
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| GB/T 7714 | Xie, Mengmeng , Zhou, Chuixiang . Signed Circuit Cover of Bridgeless Signed Graphs [J]. | GRAPHS AND COMBINATORICS , 2020 , 36 (5) : 1423-1443 . |
| MLA | Xie, Mengmeng 等. "Signed Circuit Cover of Bridgeless Signed Graphs" . | GRAPHS AND COMBINATORICS 36 . 5 (2020) : 1423-1443 . |
| APA | Xie, Mengmeng , Zhou, Chuixiang . Signed Circuit Cover of Bridgeless Signed Graphs . | GRAPHS AND COMBINATORICS , 2020 , 36 (5) , 1423-1443 . |
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Abstract ：
The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We prove that this conjecture is true for connected cubic graphs with a 2-factor consisting of three cycles. (C) 2020 Elsevier B.V. All rights reserved.
Keyword ：
2-factor 2-factor Cubic graphs Cubic graphs Decomposition Decomposition Spanning tree Spanning tree
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| GB/T 7714 | Xie, Mengmeng , Zhou, Chuixiang , Zhou, Shun . Decomposition of cubic graphs with a 2-factor consisting of three cycles [J]. | DISCRETE MATHEMATICS , 2020 , 343 (6) . |
| MLA | Xie, Mengmeng 等. "Decomposition of cubic graphs with a 2-factor consisting of three cycles" . | DISCRETE MATHEMATICS 343 . 6 (2020) . |
| APA | Xie, Mengmeng , Zhou, Chuixiang , Zhou, Shun . Decomposition of cubic graphs with a 2-factor consisting of three cycles . | DISCRETE MATHEMATICS , 2020 , 343 (6) . |
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Let G be a 2-edge-connected simple graph on n vertices, n >= 3. It is known that if G satisfies that d(x) >= n/2 for every vertex x is an element of V(G), then G has a nowhere-zero 3-flow with several exceptions. In this paper, we prove that with ten exceptions, all graphs with at most two vertices of degree less than n/2 have nowhere-zero 3-flows. More precisely, if G is a 2-edge-connected graph on n vertices, n >= 3, in which at most two vertices have degree less than n/2, then G has no nowhere-zero 3-flow if and only if G is one of ten completely described graphs.
Keyword ：
minimum degree minimum degree nowhere-zero 3-flow nowhere-zero 3-flow
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| GB/T 7714 | Zhou, Chuixiang . Minimum degree and nowhere-zero 3-flows [J]. | ARS COMBINATORIA , 2013 , 110 : 161-178 . |
| MLA | Zhou, Chuixiang . "Minimum degree and nowhere-zero 3-flows" . | ARS COMBINATORIA 110 (2013) : 161-178 . |
| APA | Zhou, Chuixiang . Minimum degree and nowhere-zero 3-flows . | ARS COMBINATORIA , 2013 , 110 , 161-178 . |
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A balanced bipartition of a graph G is a partition of V (G) into two subsets V-1 and V-2, which differ in size by at most 1. The minimum balanced bipartition problem asks for a balanced bipartition V-1. V-2 of a graph minimizing e(V-1 V-2), where e(V-1, V-2) is the number of edges joining V-1 and V-2. We present a tight upper bound on the minimum of e(V-1, V-2), giving one answer to a question of Bollobas and Scott. We prove that every connected triangle-free plane graph G of order at least 3 has a balanced bipartition V-1, V-2 with e(V-1, V-2) <= vertical bar V(G)vertical bar - 2, and we show that K-1.3, K-3.3 - e, and K-2,K-n, with n >= 1, are precisely the extremal graphs. We also show that every plane graph G without separating triangles has a balanced bipartition V-1. V-2 such that e(V-1, V-2) <= vertical bar V(G)vertical bar + 1. (C) 2011 Elsevier B.V. All rights reserved.
Keyword ：
Balanced bipartition Balanced bipartition Plane graph Plane graph Upper bound Upper bound
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| GB/T 7714 | Fan, Genghua , Xu, Baogang , Yu, Xingxing et al. Upper bounds on minimum balanced bipartitions [J]. | DISCRETE MATHEMATICS , 2012 , 312 (6) : 1077-1083 . |
| MLA | Fan, Genghua et al. "Upper bounds on minimum balanced bipartitions" . | DISCRETE MATHEMATICS 312 . 6 (2012) : 1077-1083 . |
| APA | Fan, Genghua , Xu, Baogang , Yu, Xingxing , Zhou, Chuixiang . Upper bounds on minimum balanced bipartitions . | DISCRETE MATHEMATICS , 2012 , 312 (6) , 1077-1083 . |
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Abstract ：
Bondy和Vince曾证明最小度不小于3的图包含两个长度相差为1或者2的圈,这个结果回答了Erd(o|¨)s提出的问题.H(o|¨)ggkvist和scott证明了除K_4外,所有的3-正则图都包含两个长度相差2的圈.通过不同的方法,我们得到了下面的结论:除了每个端块都是K_4的图外,所有最小度不小于3的图都包含两个长度相差2的圈.
Keyword ：
圈 圈 最小度 最小度 长度 长度
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| GB/T 7714 | 周垂香 . 最小度不小于3的图的圈长问题(英文) [J]. | 数学研究 , 2011 , 44 (03) : 270-282 . |
| MLA | 周垂香 . "最小度不小于3的图的圈长问题(英文)" . | 数学研究 44 . 03 (2011) : 270-282 . |
| APA | 周垂香 . 最小度不小于3的图的圈长问题(英文) . | 数学研究 , 2011 , 44 (03) , 270-282 . |
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Abstract ：
Bondy和Vince曾证明最小度不小于3的图包含两个长度相差为1或者2的圈，这个结果回答了ErdSs提出的问题．Haggkvist和scott证明了除肠外，所有的3-正则图都包含两个长度相差2的圈．通过不同的方法，我们得到了下面的结论：除了每个端块都是硒的图外，所有最小度不小于3的图都包含两个长度相差2的圈．
Keyword ：
圈 圈 最小度 最小度 长度 长度
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| GB/T 7714 | 周垂香 . 最小度不小于3的图的圈长问题 [J]. | 数学研究 , 2011 , 44 (3) : 270-282 . |
| MLA | 周垂香 . "最小度不小于3的图的圈长问题" . | 数学研究 44 . 3 (2011) : 270-282 . |
| APA | 周垂香 . 最小度不小于3的图的圈长问题 . | 数学研究 , 2011 , 44 (3) , 270-282 . |
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