Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, ...Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111： 416-419]. We also generalizes an early result of M. Nanasiova and M. Skoviera [J. Algebraic Combin., 2009, 30： 103-110].展开更多
The base graph of a simple matroid M = （E, A） is the graph G such that V（G） = A and E（G） = {BB＇： B, B＇ B, [B ／ B＇｜ = 1}, where the same notation is used for the vertices of G and the bases of M. It is prov...The base graph of a simple matroid M = （E, A） is the graph G such that V（G） = A and E（G） = {BB＇： B, B＇ B, [B ／ B＇｜ = 1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connected simple matroid M is Z3-connected if ｜V（G）｜ ≥5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if IV（G）[ =4. Furthermore, if for every connected component Ei （ i≥ 2） of M, the matroid base graph Gi of Mi=MIEi has IV（Gi）｜≥5, then G is Z3-connected which also implies that G admits nowhere-zero 3-flow immediately.展开更多
Let G be a bridgeless graph and C be a circuit in G.To find a shorter circuit cover of G,Fan proposed a conjecture that if G/C admits a nowhere-zero 4-flow,then G admits a 4-flow(D,f)such that E(G)E(C)?supp(f)and|supp...Let G be a bridgeless graph and C be a circuit in G.To find a shorter circuit cover of G,Fan proposed a conjecture that if G/C admits a nowhere-zero 4-flow,then G admits a 4-flow(D,f)such that E(G)E(C)?supp(f)and|supp(f)∩E(C)|>3/4|E(C)|,and showed that the conjecture holds if|E(C)|≤19[Combinatorica,37,1097–1112(2017)].In this paper,we prove that the conjecture holds if|E(C)|≤27.展开更多
A graph G is k-triangular if each of its edge is contained in at least k triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph G is a sequen...A graph G is k-triangular if each of its edge is contained in at least k triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph G is a sequence of distinct triangles T1T2%…Tk in G such that for 1 〈 i 〈 k - 1, IE（Ti）∩E（Ti＋1）1= 1 and E（Ti） n E（Tj）=φ if j 〉 i＋1. Two edges e, e＇∈ E（G） are triangularly connected if there is a triangle-path T1, T2,... , Tk in G such that e ∈ E（T1） and er ∈ E（Tk）. Two edges e, e＇ ∈E（G） are equivalent if they are the same, parallel or triangularly connected. It is easy to see that this is an equivalent relation. Each equivalent class is called a triangularly connected component. In this paper, we prove that every 4-edge-connected triangular graph G is Z3-connected, unless it has a triangularly connected component which is not Z3-connected but admits a nowhere-zero 3-flow.展开更多
Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E（G）, define d（e） = d（u） ＋ d（v）. Let F denote the set of all simple 2-edge-connected graphs on n ） 4 vertices such that G∈ F if ...Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E（G）, define d（e） = d（u） ＋ d（v）. Let F denote the set of all simple 2-edge-connected graphs on n ） 4 vertices such that G∈ F if and only if d（e） ＋ d（e′） ≥ 2n for every pair of independent edges e, e′ of G. We prove in this paper that for each G ∈ F, G is not Z3-connected if and only if G is one of K2,n-2, K3,n-3, K^＋2,n-2,K^＋ 3,n-3 or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 20]0, 310： 3390-3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308： 6233-6240].展开更多
基金Acknowledgements The first author was supported by the Natural Science Foundation of China （Grant No. 11301254）, the Natural Science Foundation of Henan Province （Grant No. 132300410313）, and the Natural Science Foundation of Education Bureau of Henan Province （Grant No. 13A110800）. The second author was supported by the National Natural Science Foundation of China （Grant No. 11171129） and the Doctoral Fund of Ministry of Education of China （Grant No. 20130144110001）.
文摘Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111： 416-419]. We also generalizes an early result of M. Nanasiova and M. Skoviera [J. Algebraic Combin., 2009, 30： 103-110].
文摘The base graph of a simple matroid M = （E, A） is the graph G such that V（G） = A and E（G） = {BB＇： B, B＇ B, [B ／ B＇｜ = 1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connected simple matroid M is Z3-connected if ｜V（G）｜ ≥5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if IV（G）[ =4. Furthermore, if for every connected component Ei （ i≥ 2） of M, the matroid base graph Gi of Mi=MIEi has IV（Gi）｜≥5, then G is Z3-connected which also implies that G admits nowhere-zero 3-flow immediately.
基金Supported by National Natural Science Foundation of China(Grant Nos.11871397,12071370 and U1803263)the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2020JM-083)。
文摘Let G be a bridgeless graph and C be a circuit in G.To find a shorter circuit cover of G,Fan proposed a conjecture that if G/C admits a nowhere-zero 4-flow,then G admits a 4-flow(D,f)such that E(G)E(C)?supp(f)and|supp(f)∩E(C)|>3/4|E(C)|,and showed that the conjecture holds if|E(C)|≤19[Combinatorica,37,1097–1112(2017)].In this paper,we prove that the conjecture holds if|E(C)|≤27.
文摘A graph G is k-triangular if each of its edge is contained in at least k triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph G is a sequence of distinct triangles T1T2%…Tk in G such that for 1 〈 i 〈 k - 1, IE（Ti）∩E（Ti＋1）1= 1 and E（Ti） n E（Tj）=φ if j 〉 i＋1. Two edges e, e＇∈ E（G） are triangularly connected if there is a triangle-path T1, T2,... , Tk in G such that e ∈ E（T1） and er ∈ E（Tk）. Two edges e, e＇ ∈E（G） are equivalent if they are the same, parallel or triangularly connected. It is easy to see that this is an equivalent relation. Each equivalent class is called a triangularly connected component. In this paper, we prove that every 4-edge-connected triangular graph G is Z3-connected, unless it has a triangularly connected component which is not Z3-connected but admits a nowhere-zero 3-flow.
基金Acknowledgements The first author was supported by the Excellent Doctorial Dissertation Cultivation Grant from Huazhong Normal University （2013YBYB42）. The second author was supported in part by the National Natural Science Foundation of China （Grant No. 11171129） and the Doctoral Fund of Ministry of Education of China （Grant No. 20130144110001）.
文摘Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E（G）, define d（e） = d（u） ＋ d（v）. Let F denote the set of all simple 2-edge-connected graphs on n ） 4 vertices such that G∈ F if and only if d（e） ＋ d（e′） ≥ 2n for every pair of independent edges e, e′ of G. We prove in this paper that for each G ∈ F, G is not Z3-connected if and only if G is one of K2,n-2, K3,n-3, K^＋2,n-2,K^＋ 3,n-3 or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 20]0, 310： 3390-3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308： 6233-6240].
