Let G be a graph, the square graph G 2 of G is a graph satisfying V(G 2)=V(G) and E(G 2)=E(G)∪｛uv: dist G(u, v)=2｝ . In this paper, we use the technique of vertex insertion on l -connected ( l=k or k...Let G be a graph, the square graph G 2 of G is a graph satisfying V(G 2)=V(G) and E(G 2)=E(G)∪｛uv: dist G(u, v)=2｝ . In this paper, we use the technique of vertex insertion on l -connected ( l=k or k+1, k≥2 ) claw-free graphs to provide a unified proof for G to be Hamiltonian, 1 -Hamiltonian or Hamiltonian-connected. The sufficient conditions are expressed by the inequality concerning ∑ k i=0N(Y i) and n(Y) in G for each independent set Y=｛y 0, y 1, …, y k｝ of the square graph of G , where b ( 0<b<k+1 ) is an integer, Y i=｛y i, y i-1, …, y i-(b-1)｝Y for i∈｛0, 1, …, k｝ , where subscriptions of y j s will be taken modulo k+1 , and n(Y)=｛v∈ V(G): dist (v, Y)≤ 2｝ .展开更多
A graph G is called quasi-claw-free if it satisfies the property： d（x, y） = 2 there exists a vertex u ∈ N（x） ∩ N（y） such that N[u] N[x] ∪ N[y]. In this paper, we show that every 2-connected quasi-claw-free g...A graph G is called quasi-claw-free if it satisfies the property： d（x, y） = 2 there exists a vertex u ∈ N（x） ∩ N（y） such that N[u] N[x] ∪ N[y]. In this paper, we show that every 2-connected quasi-claw-free graph of order n with G F contains a cycle of length at least min{3δ ＋ 2, n}, where F is a family of graphs.展开更多
A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has th...A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has the single k-cycle property if every edge of G,which does not lie in a triangle,lies in a cycle C of order at most k such that C has at least「｜V（C） /2」 edges which do not lie in a triangle,and they are not adjacent.In this paper,we show that every hourglass-free claw-free graph G of δ（G） ≥3 with the single 7-cycle property is Hamiltonian and is best possible;we also show that every claw-free graph G of δ（G） ≥3 with the hourglass property and with single 6-cycle property is Hamiltonian.展开更多
Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u.For any UV(G),let N(U)=∪_~u∈U N(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgra...Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u.For any UV(G),let N(U)=∪_~u∈U N(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K_~1,3 .One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al.: Let G be a 2-connected claw-free graph of order n,and d(u)+d(v)+d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that,for any three positive integers s,t and w,such that if G is a (s+t+w-1)-connected claw-free graph of order n,and d(S)+d(T)+d(W)>n-(s+t+w) for every three disjoint independent vertex sets S,T,W with |S|=s,|T|=t,|W|=w,and S∪T∪W is also independent,then G is Hamiltonian.Other related results are obtained too.展开更多
A spanning tree with no more than 3 leaves is called a spanning 3-ended tree. In this paper, we prove that if G is a k-connected （k≥ 2） almost claw-free graph of order n and σk＋3（G） ≥ n ＋ k ＋ 2, then G conta...A spanning tree with no more than 3 leaves is called a spanning 3-ended tree. In this paper, we prove that if G is a k-connected （k≥ 2） almost claw-free graph of order n and σk＋3（G） ≥ n ＋ k ＋ 2, then G contains a spanning 3-ended tree, where σk（G） = min{∑es deg（v） ： S is an independent set of G with ｜S｜ = k}.展开更多
A known result by Jackson Bill is that every 2-connected k-regular graph on at most 3k vertices is Hamiltonian. In this paper,it is proved that every 2-connected k-regular claw-free graph on at most 5k(k≥10)vertices ...A known result by Jackson Bill is that every 2-connected k-regular graph on at most 3k vertices is Hamiltonian. In this paper,it is proved that every 2-connected k-regular claw-free graph on at most 5k(k≥10)vertices is Hamiltonian. Moreover, the bound 5k is best possible. A counterexample of a 2-connected k-regular claw-free non-Hamiltonian graph on 5k+1 vertices is given, and it is conjectured that every 3-connected k-regular claw-free graph on at most 12k-7 vertices is Hamiltonian.展开更多
The induced matching cover number of a graph G without isolated vertices, denoted by imc(G),is the minimum integer k such that G has k induced matchings {M1,M2,···,Mk}such that,V(M1)∪V(M2)∪··...The induced matching cover number of a graph G without isolated vertices, denoted by imc(G),is the minimum integer k such that G has k induced matchings {M1,M2,···,Mk}such that,V(M1)∪V(M2)∪···∪V(Mk)covers V(G).This paper shows that,if G is a 3-regular claw-free graph,then imc(G)∈{2,3}.展开更多
In this paper we consider a property of claw-free graphs.We show that if d(u)+ d(v)≥ν(G)+2k+3,for every two nonadjacent vertices u and v,then G is 2k-vertex-deletable IM-extendable,whereν(G)=|V(G)|.And the bound is...In this paper we consider a property of claw-free graphs.We show that if d(u)+ d(v)≥ν(G)+2k+3,for every two nonadjacent vertices u and v,then G is 2k-vertex-deletable IM-extendable,whereν(G)=|V(G)|.And the bound is tight.展开更多
A graph is called claw-free if it does not contain a claw as its induced subgraph.In this paper, we prove the following results:1)If G is a 2-connected claw-free graph on n vertices,then for any vertex v and any two d...A graph is called claw-free if it does not contain a claw as its induced subgraph.In this paper, we prove the following results:1)If G is a 2-connected claw-free graph on n vertices,then for any vertex v and any two distinct vertices x and y in V(G)-{v},G has a path containing v and all neighbors of v and connecting x and y;2) Let C be the longest cycle in a 3-connected claw-free graph G and H a component of G-C,and if H is connected but not 2-connected,then there exist nonadjacent vertices u and v in H such that |V(C)|≥(3(d(u)+)d(v))-2.展开更多
A coloring of edges of a graph G is injective if for any two distinct edges e1 and e2,the coloring of e1 and e2 are distinct if they are at distance 2 in G or in a common 3-cycle.The injective chromatic index of G is ...A coloring of edges of a graph G is injective if for any two distinct edges e1 and e2,the coloring of e1 and e2 are distinct if they are at distance 2 in G or in a common 3-cycle.The injective chromatic index of G is the minimum number of colors needed for an injective edge coloring of G.It was conjectured that the injective chromatic index of any subcubic graph is at most 6.In this paper,we partially confirm this conjecture by showing that the injective chromatic index of any claw-free subcubic graph is less than or equal to 6.The bound 6 is tight and our proof implies a linear-time algorithm for finding an injective edge coloring using at most 6 colors for such graphs.展开更多
Let G be a graph and C be an arbitrary even cycle of G.The graph G is called a cycle-forced graph if G-V(C)has a unique perfect matching.When C is an arbitrary induced even cycle of G,G is called an induced-cycle-forc...Let G be a graph and C be an arbitrary even cycle of G.The graph G is called a cycle-forced graph if G-V(C)has a unique perfect matching.When C is an arbitrary induced even cycle of G,G is called an induced-cycle-forced graph.If G-V(C)has no perfect matching,G is said to be cycle-bad.This paper gives characterizations of these three type of graphs in the class of 2-connected claw-free cubic graphs.展开更多
For a graph G and an positive integer k,letσ_k(G)denote the minimum degree sum of k independent vertices of G.It has been proved that if a connected claw-free graph G satisfiesσ_(k+1)(G)≥|G|-k,then G has a spanning...For a graph G and an positive integer k,letσ_k(G)denote the minimum degree sum of k independent vertices of G.It has been proved that if a connected claw-free graph G satisfiesσ_(k+1)(G)≥|G|-k,then G has a spanning k-ended tree.We also know that the lower bound|G|-k is sharp.In this paper,we give a simple proof of this theorem by using another t-ended system.展开更多
We prove the following result: Let G be a 2 connected claw free graph of order n(n≥3) and connectivity k . If for any independent set S k+1 with cardinality k+1 , there exist u,v∈S k+1 ...We prove the following result: Let G be a 2 connected claw free graph of order n(n≥3) and connectivity k . If for any independent set S k+1 with cardinality k+1 , there exist u,v∈S k+1 , such that |N(u)∩N(v)|≥(n-2k)/4 ,then G is Hamiltonian.展开更多
In this paper it is shown that every connected claw-free graph G contains connected [a, max{a + 2, b}]-factors if it has [a, b]-factors, where a, b are integers and b ≥ a ≥ 1.
A clique-transversal set D of a graph C is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by To（G）, is the minimum cardinality of a clique- transversal set in G. In...A clique-transversal set D of a graph C is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by To（G）, is the minimum cardinality of a clique- transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.展开更多
M. Matthews and D. Sumner proved that if G is a 2-connected claw-free graph of order n, then c(G) min{2δb + 4, n}. In this paper, we prove that if G is a,2-connected claw-free graph on n venices, then c(G) min{3δ + ...M. Matthews and D. Sumner proved that if G is a 2-connected claw-free graph of order n, then c(G) min{2δb + 4, n}. In this paper, we prove that if G is a,2-connected claw-free graph on n venices, then c(G) min{3δ + 2, n} or G belongs to one exceptional class of graphs.展开更多
The degree d(H)of a subgraph H of a graph G is|u∈∪V(H)N(u)-V(H)|,where N(u)denotes the neighbor set of the vertex u of G.In this paper,we prove the following result on the condition of the degrees of subgraphs.Let G...The degree d(H)of a subgraph H of a graph G is|u∈∪V(H)N(u)-V(H)|,where N(u)denotes the neighbor set of the vertex u of G.In this paper,we prove the following result on the condition of the degrees of subgraphs.Let G be a 2-connected claw-free graph of order n with minimum degreeδ(G)≥3.If for any three non-adjacent subgraphs H1,H2,H3 that are isomorphic to K1,K1,K2,respectively,there is d(H1)+d(H2)+d(H3)≥n+3,then for each pair of vertices u,v∈G that is not a cut set,there exists a Hamilton path between u and v.展开更多
A graph is called claw-free if it contains no induced subgrapn lsomorpmc to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least （[V（G）I - 2...A graph is called claw-free if it contains no induced subgrapn lsomorpmc to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least （[V（G）I - 2）/3. At the workshop CSzC （Novy Smokovec, 1993）, Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of N （the graph obtained from a triangle by adding three disjoint pendant edges）. Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of Z1 （the graph obtained from a triangle by adding one pendant edge）. Our main result in this paper is a characterization of all graphs H such that a 2-connected claw-free graph G is Hamiltonian if eachend-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.end-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.展开更多
文摘Let G be a graph, the square graph G 2 of G is a graph satisfying V(G 2)=V(G) and E(G 2)=E(G)∪｛uv: dist G(u, v)=2｝ . In this paper, we use the technique of vertex insertion on l -connected ( l=k or k+1, k≥2 ) claw-free graphs to provide a unified proof for G to be Hamiltonian, 1 -Hamiltonian or Hamiltonian-connected. The sufficient conditions are expressed by the inequality concerning ∑ k i=0N(Y i) and n(Y) in G for each independent set Y=｛y 0, y 1, …, y k｝ of the square graph of G , where b ( 0<b<k+1 ) is an integer, Y i=｛y i, y i-1, …, y i-(b-1)｝Y for i∈｛0, 1, …, k｝ , where subscriptions of y j s will be taken modulo k+1 , and n(Y)=｛v∈ V(G): dist (v, Y)≤ 2｝ .
基金Supported by the National Natural Science Foundation of China(Grant No.60673046)
文摘A graph G is called quasi-claw-free if it satisfies the property： d（x, y） = 2 there exists a vertex u ∈ N（x） ∩ N（y） such that N[u] N[x] ∪ N[y]. In this paper, we show that every 2-connected quasi-claw-free graph of order n with G F contains a cycle of length at least min{3δ ＋ 2, n}, where F is a family of graphs.
基金Supported by the National Natural Science Foundation of China(11071016 and 11171129)the Beijing Natural Science Foundation(1102015)
文摘A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has the single k-cycle property if every edge of G,which does not lie in a triangle,lies in a cycle C of order at most k such that C has at least「｜V（C） /2」 edges which do not lie in a triangle,and they are not adjacent.In this paper,we show that every hourglass-free claw-free graph G of δ（G） ≥3 with the single 7-cycle property is Hamiltonian and is best possible;we also show that every claw-free graph G of δ（G） ≥3 with the hourglass property and with single 6-cycle property is Hamiltonian.
文摘Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u.For any UV(G),let N(U)=∪_~u∈U N(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K_~1,3 .One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al.: Let G be a 2-connected claw-free graph of order n,and d(u)+d(v)+d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that,for any three positive integers s,t and w,such that if G is a (s+t+w-1)-connected claw-free graph of order n,and d(S)+d(T)+d(W)>n-(s+t+w) for every three disjoint independent vertex sets S,T,W with |S|=s,|T|=t,|W|=w,and S∪T∪W is also independent,then G is Hamiltonian.Other related results are obtained too.
基金Supported by the National Natural Science Foundation of China,Tian Yuan Special Foundation(Grant No.11426125)by Educational Commission of Liaoning Province(Grant No.L2014239)
文摘A spanning tree with no more than 3 leaves is called a spanning 3-ended tree. In this paper, we prove that if G is a k-connected （k≥ 2） almost claw-free graph of order n and σk＋3（G） ≥ n ＋ k ＋ 2, then G contains a spanning 3-ended tree, where σk（G） = min{∑es deg（v） ： S is an independent set of G with ｜S｜ = k}.
文摘A known result by Jackson Bill is that every 2-connected k-regular graph on at most 3k vertices is Hamiltonian. In this paper,it is proved that every 2-connected k-regular claw-free graph on at most 5k(k≥10)vertices is Hamiltonian. Moreover, the bound 5k is best possible. A counterexample of a 2-connected k-regular claw-free non-Hamiltonian graph on 5k+1 vertices is given, and it is conjectured that every 3-connected k-regular claw-free graph on at most 12k-7 vertices is Hamiltonian.
基金Supported by the National Natural Science Foundation of China（10771179）
文摘The induced matching cover number of a graph G without isolated vertices, denoted by imc(G),is the minimum integer k such that G has k induced matchings {M1,M2,···,Mk}such that,V(M1)∪V(M2)∪···∪V(Mk)covers V(G).This paper shows that,if G is a 3-regular claw-free graph,then imc(G)∈{2,3}.
基金Supported by the National Natural Sciences Youth Foundation(10901144)
文摘In this paper we consider a property of claw-free graphs.We show that if d(u)+ d(v)≥ν(G)+2k+3,for every two nonadjacent vertices u and v,then G is 2k-vertex-deletable IM-extendable,whereν(G)=|V(G)|.And the bound is tight.
文摘A graph is called claw-free if it does not contain a claw as its induced subgraph.In this paper, we prove the following results:1)If G is a 2-connected claw-free graph on n vertices,then for any vertex v and any two distinct vertices x and y in V(G)-{v},G has a path containing v and all neighbors of v and connecting x and y;2) Let C be the longest cycle in a 3-connected claw-free graph G and H a component of G-C,and if H is connected but not 2-connected,then there exist nonadjacent vertices u and v in H such that |V(C)|≥(3(d(u)+)d(v))-2.
基金Supported by the National Natural Science Foundation of China(Grant No.11771080).
文摘A coloring of edges of a graph G is injective if for any two distinct edges e1 and e2,the coloring of e1 and e2 are distinct if they are at distance 2 in G or in a common 3-cycle.The injective chromatic index of G is the minimum number of colors needed for an injective edge coloring of G.It was conjectured that the injective chromatic index of any subcubic graph is at most 6.In this paper,we partially confirm this conjecture by showing that the injective chromatic index of any claw-free subcubic graph is less than or equal to 6.The bound 6 is tight and our proof implies a linear-time algorithm for finding an injective edge coloring using at most 6 colors for such graphs.
基金Supported by National Natural Science Foundation of China(Grant Nos.12171440 and 11971445)。
文摘Let G be a graph and C be an arbitrary even cycle of G.The graph G is called a cycle-forced graph if G-V(C)has a unique perfect matching.When C is an arbitrary induced even cycle of G,G is called an induced-cycle-forced graph.If G-V(C)has no perfect matching,G is said to be cycle-bad.This paper gives characterizations of these three type of graphs in the class of 2-connected claw-free cubic graphs.
文摘For a graph G and an positive integer k,letσ_k(G)denote the minimum degree sum of k independent vertices of G.It has been proved that if a connected claw-free graph G satisfiesσ_(k+1)(G)≥|G|-k,then G has a spanning k-ended tree.We also know that the lower bound|G|-k is sharp.In this paper,we give a simple proof of this theorem by using another t-ended system.
文摘We prove the following result: Let G be a 2 connected claw free graph of order n(n≥3) and connectivity k . If for any independent set S k+1 with cardinality k+1 , there exist u,v∈S k+1 , such that |N(u)∩N(v)|≥(n-2k)/4 ,then G is Hamiltonian.
基金the National Natural Science Foundation of China.
文摘In this paper it is shown that every connected claw-free graph G contains connected [a, max{a + 2, b}]-factors if it has [a, b]-factors, where a, b are integers and b ≥ a ≥ 1.
基金Supported by National Natural Science Foundation of China （Grant No. 60773078）, the PuJiang Project of Shanghai （Grant No. 09PJ1405000） and Key Disciplines of Shanghai Municipality （Grant No. $30104）
文摘A clique-transversal set D of a graph C is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by To（G）, is the minimum cardinality of a clique- transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.
文摘M. Matthews and D. Sumner proved that if G is a 2-connected claw-free graph of order n, then c(G) min{2δb + 4, n}. In this paper, we prove that if G is a,2-connected claw-free graph on n venices, then c(G) min{3δ + 2, n} or G belongs to one exceptional class of graphs.
基金supported by the National Natural Science Foundation of China(No.11871398)the Natural Science Basic Research Plan in Shaanxi Province of China(Program No.2018JM1032)+1 种基金the Fundamental Research Funds for the Central Universities(No.3102019ghjd003)the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University(No.ZZ2019031)
文摘The degree d(H)of a subgraph H of a graph G is|u∈∪V(H)N(u)-V(H)|,where N(u)denotes the neighbor set of the vertex u of G.In this paper,we prove the following result on the condition of the degrees of subgraphs.Let G be a 2-connected claw-free graph of order n with minimum degreeδ(G)≥3.If for any three non-adjacent subgraphs H1,H2,H3 that are isomorphic to K1,K1,K2,respectively,there is d(H1)+d(H2)+d(H3)≥n+3,then for each pair of vertices u,v∈G that is not a cut set,there exists a Hamilton path between u and v.
基金Supported by NSFC(Grant Nos.11271300 and 11571135)the project NEXLIZ–CZ.1.07/2.3.00/30.0038+1 种基金the project P202/12/G061 of the Czech Science Foundation and by the European Regional Development Fund(ERDF)the project NTIS-New Technologies for Information Society,European Centre of Excellence,CZ.1.05/1.1.00/02.0090
文摘A graph is called claw-free if it contains no induced subgrapn lsomorpmc to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least （[V（G）I - 2）/3. At the workshop CSzC （Novy Smokovec, 1993）, Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of N （the graph obtained from a triangle by adding three disjoint pendant edges）. Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of Z1 （the graph obtained from a triangle by adding one pendant edge）. Our main result in this paper is a characterization of all graphs H such that a 2-connected claw-free graph G is Hamiltonian if eachend-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.end-vertex of every induced copy of H in G has degree at least IV（G）I/3 ＋ 1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.
