In this work we propose a centrality measure for networks, which we refer to as Laplacian centrality, that provides a general framework for the centrality of a vertex based on the idea that the importance (or centrali...In this work we propose a centrality measure for networks, which we refer to as Laplacian centrality, that provides a general framework for the centrality of a vertex based on the idea that the importance (or centrality) of a vertex is related to the ability of the network to respond to the deactivation or removal of that vertex from the network. In particular, the Laplacian centrality of a vertex is defined as the relative drop of Laplacian energy caused by the deactivation of this vertex. The Laplacian energy of network G with?n?vertices is defined as , where ?is the eigenvalue of the Laplacian matrix of G. Other dynamics based measures such as that of Masuda and Kori and PageRank compute the importance of a node by analyzing the way paths pass through a node while our measure captures this information as well as the way these paths are “redistributed” when the node is deleted. The validity and robustness of this new measure are illustrated on two different terrorist social network data sets and 84 networks in James Moody’s Add Health in school friendship nomination data, and is compared with other standard centrality measures.展开更多
The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It ...The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.展开更多
In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and o...In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules.In the second part of the paper we consider a depth equality,called the depth formula,which has been initially introduced by Auslander and developed further by Huneke and Wiegand.As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.展开更多
This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity ...This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.展开更多
A vertex cycle cover of a digraph <i>H</i> is a collection C = {<em>C</em><sub>1</sub>, <em>C</em><sub>2</sub>, …, <em>C</em><sub><em&g...A vertex cycle cover of a digraph <i>H</i> is a collection C = {<em>C</em><sub>1</sub>, <em>C</em><sub>2</sub>, …, <em>C</em><sub><em>k</em></sub>} of directed cycles in <i>H</i> such that these directed cycles together cover all vertices in <i>H</i> and such that the arc sets of these directed cycles induce a connected subdigraph of <i>H</i>. A subdigraph <i>F</i> of a digraph <i>D</i> is a circulation if for every vertex in <i>F</i>, the indegree of <em>v</em> equals its out degree, and a spanning circulation if <i>F</i> is a cycle factor. Define <i>f</i> (<i>D</i>) to be the smallest cardinality of a vertex cycle cover of the digraph obtained from <i>D</i> by contracting all arcs in <i>F</i>, among all circulations <i>F</i> of <i>D</i>. Adigraph <i>D</i> is supereulerian if <i>D</i> has a spanning connected circulation. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> are nontrivial strong digraphs such that <em>D</em><sub>1</sub> is supereulerian and <em>D</em><sub>2</sub> has a cycle vertex cover C’ with |C’| ≤ |<em>V</em> (<em>D</em><sub>1</sub>)|, then the Cartesian product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is also supereulerian. In this paper, we prove that for strong digraphs<em> D</em><sub>1</sub> and <em>D</em><sub>2</sub>, if for some cycle factor <em>F</em><sub>1</sub> of <em>D</em><sub>1</sub>, the digraph formed from <em>D</em><sub>1</sub> by contracting arcs in F1 is hamiltonian with <i>f</i> (<i>D</i><sub>2</sub>) not bigger than |<em>V</em> (<em>D</em><sub>1</sub>)|, then the strong product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is supereulerian.展开更多
In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeg...In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an （additive） abelian group and A＊ = A - {0}. The graph G is A-connected if G has an orientation D（G） such that for every map b ： V（G） → A satisfying ∑v∈V（G）b（v） ： 0, there is a function f ： E（G） → A＊ such that for each vertex v ∈ V（G）, the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v is equal to b（v）. The group coloring of a graph arises from the dual concept of group connectivity. There have been lots of investigations on these subjects. This survey provides a summary of researches on group connectivity and group colorings of graphs. It contains the following sections. 1. Nowhere-zero Flows and Group Connectivity of Graphs 2. Complete Families and A-reductions 3. Reductions with Edge-deletions, Vertex-deletions and Vertex-splitting 4. Group Colorings as a Dual Concept of Group Connectivity 5. Brooks Theorem, Its Variations and Dual Forms 6. Planar Graphs 7. Group Connectivity of Graphs 7.1 Highly Connected Graphs and Collapsible Graphs 7.2 Degrees Conditions 7.3 Complementary Graphs 7.4 Products of Graphs 7.5 Graphs with Diameter at Most 2 7.6 Line Graphs and Claw-Free Graphs 7.7 Triangular Graphs 7.8 Claw-decompositions and All Tutte-orientations展开更多
A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3.The star chromatic indexχ'_(st)(G)of a graph G is the smallest integer k such that G h...A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3.The star chromatic indexχ'_(st)(G)of a graph G is the smallest integer k such that G has a star k-edge-coloring.The list star chromatic index ch'st(G)is defined analogously.The star edge coloring problem is known to be NP-complete,and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear.In this paper,we study,in contrast,the best linear upper bound for sparse graph classes.We show that for everyε>0 there exists a constant c(ε)such that if mad(G)<8/3-ε,then■and the coefficient 3/2 ofΔis the best possible.The proof applies a newly developed coloring extension method by assigning color sets with different sizes.展开更多
This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves(s...This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves(shock, rarefaction wave and contact discontinuity) is also introduced.展开更多
Let H=(V,E)be a hypergraph,where V is a set of vertices and E is a set of non-empty subsets of V called edges.If all edges of H have the same cardinality r,then H is an r-uniform hypergraph;if E consists of all r-subs...Let H=(V,E)be a hypergraph,where V is a set of vertices and E is a set of non-empty subsets of V called edges.If all edges of H have the same cardinality r,then H is an r-uniform hypergraph;if E consists of all r-subsets of V,then H is a complete r-uniform hypergraph,denoted by K_(n)^(r),where n=|V|.A hypergraph H′=(V′,E′)is called a subhypergraph of H=(V,E)if V′⊆V and E′⊆E.The edge-connectivity of a hypergraph H is the cardinality of a minimum edge set F⊆E such that H−F is not connected,where H−F=(V,E\F).An r-uniform hypergraph H=(V,E)is k-edge-maximal if every subhypergraph of H has edge-connectivity at most k,but for any edge e∈E(K_(n)^(r))\E(H),H+e contains at least one subhypergraph with edge-connectivity at least k+1.Let k and r be integers with k≥2 and r≥2,and let t=t(k,r)be the largest integer such that(t−1 r−1)≤k.That is,t is the integer satisfying(t−1 r−1)≤k<(t r−1).We prove that if H is an r-uniform k-edge-maximal hypergraph such that n=|V(H)|≥t,then(i)|E(H)|≤(t r)+(n−t)k,and this bound is best possible;(ii)|E(H)|≥(n−1)k−((t−1)k−(t r))[n/t],and this bound is best possible.展开更多
Let k be a positive integer.A graph G is k-weight choosable if,for any assignment L(e)of k real numbers to each e∈E(G),there is a mapping f:E(G)→R such that f(uv)∈L(uv)and∑e∈∂(u)^f(e)≠∑e∈∂(u)^f(e)for each uv∈...Let k be a positive integer.A graph G is k-weight choosable if,for any assignment L(e)of k real numbers to each e∈E(G),there is a mapping f:E(G)→R such that f(uv)∈L(uv)and∑e∈∂(u)^f(e)≠∑e∈∂(u)^f(e)for each uv∈E(G),where?(v)is the set of edges incident with v.As a strengthening of the famous 1-2-3-conjecture,Bartnicki,Grytczuk and Niwcyk[Weight choosability of graphs.J.Graph Theory,60,242–256(2009)]conjecture that every graph without isolated edge is 3-weight choosable.This conjecture is wildly open and it is even unknown whether there is a constant k such that every graph without isolated edge is k-weight choosable.In this paper,we show that every connected graph of maximum degree 4 is 4-weight choosable.展开更多
A graphG is supereulerian if G has a spanning eulerian subgraph.Boesch et al.[J.Graph Theory,1,79–84（1977）]proposed the problem of characterizing supereulerian graphs.In this paper,we prove that any 3-edge-connecte...A graphG is supereulerian if G has a spanning eulerian subgraph.Boesch et al.[J.Graph Theory,1,79–84（1977）]proposed the problem of characterizing supereulerian graphs.In this paper,we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph.This extends a former result of Catlin and Lai[J.Combin.Theory,Ser.B,66,123–139（1996）].展开更多
Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where...Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-trail.The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected.In this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥3.Applying this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.展开更多
It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero κ flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result...It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero κ flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet＇s conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.展开更多
Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A（n,k;A） are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n∑κ=0A（n,k;A）（x-Aκ）n-k=xn. （0.1） Similary, we define B（n...Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A（n,k;A） are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n∑κ=0A（n,k;A）（x-Aκ）n-k=xn. （0.1） Similary, we define B（n,k;A） to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n∑κ=0A（n,k;A）（x-Aκ）n-k=xn. （0.2） The main result of this paper is an explicit formula for B（n,k;A）, which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations （0.1） and （0.2）.展开更多
文摘In this work we propose a centrality measure for networks, which we refer to as Laplacian centrality, that provides a general framework for the centrality of a vertex based on the idea that the importance (or centrality) of a vertex is related to the ability of the network to respond to the deactivation or removal of that vertex from the network. In particular, the Laplacian centrality of a vertex is defined as the relative drop of Laplacian energy caused by the deactivation of this vertex. The Laplacian energy of network G with?n?vertices is defined as , where ?is the eigenvalue of the Laplacian matrix of G. Other dynamics based measures such as that of Masuda and Kori and PageRank compute the importance of a node by analyzing the way paths pass through a node while our measure captures this information as well as the way these paths are “redistributed” when the node is deleted. The validity and robustness of this new measure are illustrated on two different terrorist social network data sets and 84 networks in James Moody’s Add Health in school friendship nomination data, and is compared with other standard centrality measures.
文摘The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.
基金partly supported by the National Natural Science Foundation of China(Grant Nos.12271230,11761045 and 11971388)the Natural Science Foundation of Gansu Province(Grant No.21JR7RA297)。
文摘In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules.In the second part of the paper we consider a depth equality,called the depth formula,which has been initially introduced by Auslander and developed further by Huneke and Wiegand.As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.
文摘This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.
文摘A vertex cycle cover of a digraph <i>H</i> is a collection C = {<em>C</em><sub>1</sub>, <em>C</em><sub>2</sub>, …, <em>C</em><sub><em>k</em></sub>} of directed cycles in <i>H</i> such that these directed cycles together cover all vertices in <i>H</i> and such that the arc sets of these directed cycles induce a connected subdigraph of <i>H</i>. A subdigraph <i>F</i> of a digraph <i>D</i> is a circulation if for every vertex in <i>F</i>, the indegree of <em>v</em> equals its out degree, and a spanning circulation if <i>F</i> is a cycle factor. Define <i>f</i> (<i>D</i>) to be the smallest cardinality of a vertex cycle cover of the digraph obtained from <i>D</i> by contracting all arcs in <i>F</i>, among all circulations <i>F</i> of <i>D</i>. Adigraph <i>D</i> is supereulerian if <i>D</i> has a spanning connected circulation. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> are nontrivial strong digraphs such that <em>D</em><sub>1</sub> is supereulerian and <em>D</em><sub>2</sub> has a cycle vertex cover C’ with |C’| ≤ |<em>V</em> (<em>D</em><sub>1</sub>)|, then the Cartesian product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is also supereulerian. In this paper, we prove that for strong digraphs<em> D</em><sub>1</sub> and <em>D</em><sub>2</sub>, if for some cycle factor <em>F</em><sub>1</sub> of <em>D</em><sub>1</sub>, the digraph formed from <em>D</em><sub>1</sub> by contracting arcs in F1 is hamiltonian with <i>f</i> (<i>D</i><sub>2</sub>) not bigger than |<em>V</em> (<em>D</em><sub>1</sub>)|, then the strong product <em>D</em><sub>1</sub> and <em>D</em><sub>2</sub> is supereulerian.
文摘In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an （additive） abelian group and A＊ = A - {0}. The graph G is A-connected if G has an orientation D（G） such that for every map b ： V（G） → A satisfying ∑v∈V（G）b（v） ： 0, there is a function f ： E（G） → A＊ such that for each vertex v ∈ V（G）, the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v is equal to b（v）. The group coloring of a graph arises from the dual concept of group connectivity. There have been lots of investigations on these subjects. This survey provides a summary of researches on group connectivity and group colorings of graphs. It contains the following sections. 1. Nowhere-zero Flows and Group Connectivity of Graphs 2. Complete Families and A-reductions 3. Reductions with Edge-deletions, Vertex-deletions and Vertex-splitting 4. Group Colorings as a Dual Concept of Group Connectivity 5. Brooks Theorem, Its Variations and Dual Forms 6. Planar Graphs 7. Group Connectivity of Graphs 7.1 Highly Connected Graphs and Collapsible Graphs 7.2 Degrees Conditions 7.3 Complementary Graphs 7.4 Products of Graphs 7.5 Graphs with Diameter at Most 2 7.6 Line Graphs and Claw-Free Graphs 7.7 Triangular Graphs 7.8 Claw-decompositions and All Tutte-orientations
基金supported by National Natural Science Foundation of China(Grant No.11901318)the Fundamental Research Funds for the Central Universities,Nankai University(Grant No.63191425)supported by National Natural Science Foundation of China(Grant Nos.11571149 and 11971205)
文摘A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3.The star chromatic indexχ'_(st)(G)of a graph G is the smallest integer k such that G has a star k-edge-coloring.The list star chromatic index ch'st(G)is defined analogously.The star edge coloring problem is known to be NP-complete,and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear.In this paper,we study,in contrast,the best linear upper bound for sparse graph classes.We show that for everyε>0 there exists a constant c(ε)such that if mad(G)<8/3-ε,then■and the coefficient 3/2 ofΔis the best possible.The proof applies a newly developed coloring extension method by assigning color sets with different sizes.
基金supported by National Natural Science Foundation of China (Grant Nos. 11031001, 11101101 and 11421061)
文摘This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves(shock, rarefaction wave and contact discontinuity) is also introduced.
基金supported by the National Natural Science Foundation of China(Nos.11861066,11531011)Tianshan Youth Project of Xinjiang(2018Q066)。
文摘Let H=(V,E)be a hypergraph,where V is a set of vertices and E is a set of non-empty subsets of V called edges.If all edges of H have the same cardinality r,then H is an r-uniform hypergraph;if E consists of all r-subsets of V,then H is a complete r-uniform hypergraph,denoted by K_(n)^(r),where n=|V|.A hypergraph H′=(V′,E′)is called a subhypergraph of H=(V,E)if V′⊆V and E′⊆E.The edge-connectivity of a hypergraph H is the cardinality of a minimum edge set F⊆E such that H−F is not connected,where H−F=(V,E\F).An r-uniform hypergraph H=(V,E)is k-edge-maximal if every subhypergraph of H has edge-connectivity at most k,but for any edge e∈E(K_(n)^(r))\E(H),H+e contains at least one subhypergraph with edge-connectivity at least k+1.Let k and r be integers with k≥2 and r≥2,and let t=t(k,r)be the largest integer such that(t−1 r−1)≤k.That is,t is the integer satisfying(t−1 r−1)≤k<(t r−1).We prove that if H is an r-uniform k-edge-maximal hypergraph such that n=|V(H)|≥t,then(i)|E(H)|≤(t r)+(n−t)k,and this bound is best possible;(ii)|E(H)|≥(n−1)k−((t−1)k−(t r))[n/t],and this bound is best possible.
基金Supported by National Natural Science Foundation of China(Grant Nos.11871397 and 11971205)the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2020JM-083)the Fundamental Research Funds for the Central Universities(Grant No.3102019ghjd003)。
文摘Let k be a positive integer.A graph G is k-weight choosable if,for any assignment L(e)of k real numbers to each e∈E(G),there is a mapping f:E(G)→R such that f(uv)∈L(uv)and∑e∈∂(u)^f(e)≠∑e∈∂(u)^f(e)for each uv∈E(G),where?(v)is the set of edges incident with v.As a strengthening of the famous 1-2-3-conjecture,Bartnicki,Grytczuk and Niwcyk[Weight choosability of graphs.J.Graph Theory,60,242–256(2009)]conjecture that every graph without isolated edge is 3-weight choosable.This conjecture is wildly open and it is even unknown whether there is a constant k such that every graph without isolated edge is k-weight choosable.In this paper,we show that every connected graph of maximum degree 4 is 4-weight choosable.
基金Supported by National Natural Science Foundation of China(Grant No.11001287)Science Foundation Chongqing Education Committee(Grant Nos.KJ100725 and KJ120731)
文摘A graphG is supereulerian if G has a spanning eulerian subgraph.Boesch et al.[J.Graph Theory,1,79–84（1977）]proposed the problem of characterizing supereulerian graphs.In this paper,we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph.This extends a former result of Catlin and Lai[J.Combin.Theory,Ser.B,66,123–139（1996）].
基金This paper is supported by the National Natural Science Foundation of China(Nos.11771039,11971054)Fundamental Research Funds for the Central Universities of China(No.2015JBM107)the 111 Project of China(No.B16002)。
文摘Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-trail.The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected.In this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥3.Applying this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.
基金Supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China Project(Grant No.10XNB054)
文摘It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero κ flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet＇s conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.
文摘Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A（n,k;A） are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n∑κ=0A（n,k;A）（x-Aκ）n-k=xn. （0.1） Similary, we define B（n,k;A） to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n∑κ=0A（n,k;A）（x-Aκ）n-k=xn. （0.2） The main result of this paper is an explicit formula for B（n,k;A）, which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations （0.1） and （0.2）.
