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学者姓名：范更华
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Abstract ：
Gallai’s conjecture asserts that every connected graph on n vertices can be decomposed into at most n+12 paths. The E-subgraph of a graph G, denoted by Ge, is the subgraph induced by the vertices of even degree in G. A triangle pair is a graph consisting of two triangles with exactly one vertex in common. In this paper, it is proved that Gallai’s conjecture is true for graphs G in which Ge contains no triangle pair and each block of Ge has maximum degree at most 3. © 2022, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature.
Keyword ：
Decomposition Decomposition Gallai’s conjecture Gallai’s conjecture Path Path Triangle Triangle
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| GB/T 7714 | Fan, G.-H. , Hou, J.-F. , Zhou, C.-X. . Gallai’s Conjecture on Path Decompositions [J]. | Journal of the Operations Research Society of China , 2023 , 11 (3) : 439-449 . |
| MLA | Fan, G.-H. 等. "Gallai’s Conjecture on Path Decompositions" . | Journal of the Operations Research Society of China 11 . 3 (2023) : 439-449 . |
| APA | Fan, G.-H. , Hou, J.-F. , Zhou, C.-X. . Gallai’s Conjecture on Path Decompositions . | Journal of the Operations Research Society of China , 2023 , 11 (3) , 439-449 . |
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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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Let G be a signed graph and F a set of signed circuits in G. For an edge e is an element of E(G), F(e) denotes the number of signed circuits in F that contain e. F is called a circuit-cover of G if F(e) > 0 for each e is an element of E(G), and a circuit k-cover of G if F(e) = k for each e is an element of E(G). G is coverable if it has a circuit-cover. The existence of a circuit-cover in G is equivalent to the existence of a nowhere-zero flow in G. For a coverable signed graph G, it is proved in this paper that if each maximal 2-edge-connected subgraph of G is eulerian, then G has a circuit 6-cover, consisting of four circuit-covers of G, and as an immediate consequence, G has a circuit-cover of length at most 3/2 vertical bar E(G)vertical bar, generalizing a known result on signed eulerian graphs. New results on circuit k-covers are obtained and applied to estimating bounds on the lengths of shortest circuit-covers of signed graphs. (c) 2021 Elsevier B.V. All rights reserved.
Keyword ：
Circuit k-cover Circuit k-cover Short circuit cover Short circuit cover Signed eulerian graph Signed eulerian graph Signed graph Signed graph
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| GB/T 7714 | Chen, Jing , Fan, Genghua . Circuit k-covers of signed graphs [J]. | DISCRETE APPLIED MATHEMATICS , 2021 , 294 : 41-54 . |
| MLA | Chen, Jing 等. "Circuit k-covers of signed graphs" . | DISCRETE APPLIED MATHEMATICS 294 (2021) : 41-54 . |
| APA | Chen, Jing , Fan, Genghua . Circuit k-covers of signed graphs . | DISCRETE APPLIED MATHEMATICS , 2021 , 294 , 41-54 . |
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A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.
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| GB/T 7714 | Bao, Bo , Chen, Rong , Fan, Genghua . Circuit Covers of Signed Eulerian Graphs [J]. | ELECTRONIC JOURNAL OF COMBINATORICS , 2021 , 28 (1) . |
| MLA | Bao, Bo 等. "Circuit Covers of Signed Eulerian Graphs" . | ELECTRONIC JOURNAL OF COMBINATORICS 28 . 1 (2021) . |
| APA | Bao, Bo , Chen, Rong , Fan, Genghua . Circuit Covers of Signed Eulerian Graphs . | ELECTRONIC JOURNAL OF COMBINATORICS , 2021 , 28 (1) . |
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Gallai's conjecture asserts that a connected graph G on n vertices can be decomposed into [n/2] paths. In this paper, we prove that a connected graph with maximum degree 6, in which the vertices of degree 6 form an independent set, can be decomposed into n/2 paths, unless it is K-3, K-5 and K-5(-). (C) 2020 Elsevier B.V. All rights reserved.
Keyword ：
Decomposition Decomposition Gallai's conjecture Gallai's conjecture Path Path
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| GB/T 7714 | Chu, Yanan , Fan, Genghua , Liu, Qinghai . On Gallai's conjecture for graphs with maximum degree 6 [J]. | DISCRETE MATHEMATICS , 2021 , 344 (2) . |
| MLA | Chu, Yanan 等. "On Gallai's conjecture for graphs with maximum degree 6" . | DISCRETE MATHEMATICS 344 . 2 (2021) . |
| APA | Chu, Yanan , Fan, Genghua , Liu, Qinghai . On Gallai's conjecture for graphs with maximum degree 6 . | DISCRETE MATHEMATICS , 2021 , 344 (2) . |
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Let T be a tree with m edges. It was conjectured that every m-regular bipartite graph can be decomposed into edge-disjoint copies of T. In this paper, we prove that every 6-regular bipartite graph can be decomposed into edge-disjoint paths with 6 edges. As a consequence, every 6-regular bipartite graph on n vertices can be decomposed into n/2 paths, which is related to the well-known Gallai's Conjecture: every connected graph on n vertices can be decomposed into at most n+1/2 paths.
Keyword ：
Decomposition Decomposition Path Path Regular graph Regular graph
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| GB/T 7714 | Chu, Yanan , Fan, Genghua , Zhou, Chuixiang . Decompositions of 6-Regular Bipartite Graphs into Paths of Length Six [J]. | GRAPHS AND COMBINATORICS , 2020 , 37 (1) : 263-269 . |
| MLA | Chu, Yanan 等. "Decompositions of 6-Regular Bipartite Graphs into Paths of Length Six" . | GRAPHS AND COMBINATORICS 37 . 1 (2020) : 263-269 . |
| APA | Chu, Yanan , Fan, Genghua , Zhou, Chuixiang . Decompositions of 6-Regular Bipartite Graphs into Paths of Length Six . | GRAPHS AND COMBINATORICS , 2020 , 37 (1) , 263-269 . |
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Let M be a loopless matroid on E with rank function r(m). Let beta(M) = max(empty set not equal X subset of E) vertical bar X vertical bar/R-M(X) and phi(M) = Minr(M)(X)= 0 is an integer and 0 <= epsilon < 1, then E can be partitioned into k + 1 independent sets with one of size at most epsilon . r(M)(E). If ca(M) = k + epsilon, then M has k + 1 disjoint independent sets such that k are bases and the other has size at least epsilon . r(M)(E). We apply these results to truncations of cycle matroids to refine graph-theoretic results due to Chen, Koh, and Peng in 1993 and to Chen and Lai in 1996. (C) 2018 Elsevier Ltd. All rights reserved.
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| GB/T 7714 | Fan, Genghua , Jiang, Hongbi , Li, Ping et al. Extensions of matroid covering and packing [J]. | EUROPEAN JOURNAL OF COMBINATORICS , 2019 , 76 : 117-122 . |
| MLA | Fan, Genghua et al. "Extensions of matroid covering and packing" . | EUROPEAN JOURNAL OF COMBINATORICS 76 (2019) : 117-122 . |
| APA | Fan, Genghua , Jiang, Hongbi , Li, Ping , West, Douglas B. , Yang, Daqing , Zhu, Xuding . Extensions of matroid covering and packing . | EUROPEAN JOURNAL OF COMBINATORICS , 2019 , 76 , 117-122 . |
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A signed graph G is a graph associated with a mapping (sigma: E(G) -> {+1, -1). An edge e is an element of E(G) is positive if sigma(e) = 1 and negative if sigma(e) = -1. A circuit in G is balanced if it contains an even number of negative edges, and unbalanced otherwise. A barbell consists of two unbalanced circuits joined by a path. A signed circuit of G is either a balanced circuit or a barbell. A signed graph is coverable if each edge is contained in some signed circuit. An oriented signed" graph (bidirected graph) has a nowhere-zero integer flow if and only if it is coverable. A signed circuit cover of G is a collection.F of signed circuits in G such that each edge e is an element of E(G) is contained in at least one signed circuit of.T; The length of.F is the sum of the lengths of the signed circuits in it. The minimum length of a signed circuit cover of G is denoted by scc(G). The first nontrivial bound on scc(G) was established by Mgajova et al., who proved that scc(G) <= 11|E(G)| for every coverable signed graph G, which was recently improved by Cheng et al. to scc(G) <= 14/3 |E(G)|. In this paper, we prove that scc(G) <= 25/6 |E(G)| for every coverable signed graph G. (C) 2017 Elsevier B.V. All rights reserved.
Keyword ：
Circuit cover Circuit cover Minimum circuit cover Minimum circuit cover Signed circuit Signed circuit Signed graph Signed graph
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| GB/T 7714 | Chen, Jing , Fan, Genghua . Short signed circuit covers of signed graphs [J]. | DISCRETE APPLIED MATHEMATICS , 2018 , 235 : 51-58 . |
| MLA | Chen, Jing et al. "Short signed circuit covers of signed graphs" . | DISCRETE APPLIED MATHEMATICS 235 (2018) : 51-58 . |
| APA | Chen, Jing , Fan, Genghua . Short signed circuit covers of signed graphs . | DISCRETE APPLIED MATHEMATICS , 2018 , 235 , 51-58 . |
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A spanning subgraph F of a graph G is called an even factor of G if each vertex of F has even degree at least 2 in F. It was conjectured that if a graph G has an even factor, then it has an even factor F with , where is the set of vertices of degree 2 in G. We note that the conjecture is false if G is a triangle. In this paper, we confirm the conjecture for all graphs on at least 4 vertices, and moreover, we prove that if for every even factor H of G, then every maximum even factor of G is a 2-factor consisting of even circuits.
Keyword ：
2-factor 2-factor Even factor Even factor Extremal graph Extremal graph Spanning subgraph Spanning subgraph
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| GB/T 7714 | Chen, Jing , Fan, Genghua . Large even factors of graphs [J]. | JOURNAL OF COMBINATORIAL OPTIMIZATION , 2018 , 35 (1) : 162-169 . |
| MLA | Chen, Jing et al. "Large even factors of graphs" . | JOURNAL OF COMBINATORIAL OPTIMIZATION 35 . 1 (2018) : 162-169 . |
| APA | Chen, Jing , Fan, Genghua . Large even factors of graphs . | JOURNAL OF COMBINATORIAL OPTIMIZATION , 2018 , 35 (1) , 162-169 . |
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Bollobas and Scott (Random Struct. Alg. 21 (2002) 414-430) asked for conditions that guarantee a bisection of a graph with m edges in which each class has at most (1/4 + o(1)) m edges. We demonstrate that cycles of length 4 play an important role for this question. Let G be a graph with m edges, minimum degree delta, and containing no cycle of length 4. We show that if (i) G is 2-connected, or (ii) delta >= 3, or (iii) delta >= 2 and the girth of G is at least 5, then G admits a bisection in which each class has at most (1/4+ o(1)) m edges. We show that each of these conditions are best possible. On the other hand, a construction by Alon, Bollobas, Krivelevich and Sudakov shows that for infinitely many m there exists a graph with m edges and girth at least 5 for which any bisection has at least (1/4 - o(1)) m edges in one of the two classes.
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| GB/T 7714 | Fan, Genghua , Hou, Jianfeng , Yu, Xingxing . Bisections of Graphs Without Short Cycles [J]. | COMBINATORICS PROBABILITY & COMPUTING , 2018 , 27 (1) : 44-59 . |
| MLA | Fan, Genghua et al. "Bisections of Graphs Without Short Cycles" . | COMBINATORICS PROBABILITY & COMPUTING 27 . 1 (2018) : 44-59 . |
| APA | Fan, Genghua , Hou, Jianfeng , Yu, Xingxing . Bisections of Graphs Without Short Cycles . | COMBINATORICS PROBABILITY & COMPUTING , 2018 , 27 (1) , 44-59 . |
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