A path factor of G is a spanning subgraph of G such that its each component is a path.A path factor is called a P≥n-factor if its each component admits at least n vertices.A graph G is called P≥n-factor covered if G...A path factor of G is a spanning subgraph of G such that its each component is a path.A path factor is called a P≥n-factor if its each component admits at least n vertices.A graph G is called P≥n-factor covered if G admits a P≥n-factor containing e for any e∈E(G),which is defined by[Discrete Mathematics,309,2067-2076(2009)].We first define the concept of a(P≥n,k)-factor-critical covered graph,namely,a graph G is called(P≥n,k)-factor-critical covered if G-D is P≥n-factor covered for any D⊆V(G)with|D|=k.In this paper,we verify that(i)a graph G withκ(G)≥k+1 is(P≥2,k)-factor-critical covered if bind(G)>2+k/3;(ii)a graph G with|V(G)|≥k+3 andκ(G)≥k+1 is(P≥3,k)-factor-critical covered if bind(G)≥4+k/3.展开更多
Let a,b,k,r be nonnegative integers with 1 ≤ a ≤b and r ≥ 2. Let G be a graph of order n with n 〉 （a＋b）（r（a＋b）-2）＋ak/a. In this paper, we first show a characterization for all fractional （a, b, k）-criti...Let a,b,k,r be nonnegative integers with 1 ≤ a ≤b and r ≥ 2. Let G be a graph of order n with n 〉 （a＋b）（r（a＋b）-2）＋ak/a. In this paper, we first show a characterization for all fractional （a, b, k）-critical graphs. Then using the result, we prove that G is all fractional （a, b, k）-critical if δ（G） ≥ （r-1）b2/a ＋k and ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn＋ak/a＋b for any independent subset {xl, x2, .., xr} in G. Furthermore, it is shown that the lower bound on the condition ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn=ak/ a＋b is best possible in some sense, and it is an extension of Lu＇s previous result.展开更多
LetG be a graph,and k≥2 be a positive integer.A graph G is fractional independentset-deletable k-factor-critical（in short,fractional ID-k-factor-critical）,if G I has a fractional k-factor for every independent set ...LetG be a graph,and k≥2 be a positive integer.A graph G is fractional independentset-deletable k-factor-critical（in short,fractional ID-k-factor-critical）,if G I has a fractional k-factor for every independent set I of G.The binding number bind（G）of a graph G is defined as bind（G）=min｜NG（X）｜｜X｜：=X V（G）,NG（X）=V（G）.In this paper,it is proved that a graph G is fractional ID-k-factor-critical if n≥6k 9 and bind（G）〉（3k 1）（n 1）kn 2k＋2.展开更多
基金Supported by Six Big Talent Peak of Jiangsu Province(Grant No.JY–022)333 Project of Jiangsu Provincethe National Natural Science Foundation of China(Grant No.11371009)。
文摘A path factor of G is a spanning subgraph of G such that its each component is a path.A path factor is called a P≥n-factor if its each component admits at least n vertices.A graph G is called P≥n-factor covered if G admits a P≥n-factor containing e for any e∈E(G),which is defined by[Discrete Mathematics,309,2067-2076(2009)].We first define the concept of a(P≥n,k)-factor-critical covered graph,namely,a graph G is called(P≥n,k)-factor-critical covered if G-D is P≥n-factor covered for any D⊆V(G)with|D|=k.In this paper,we verify that(i)a graph G withκ(G)≥k+1 is(P≥2,k)-factor-critical covered if bind(G)>2+k/3;(ii)a graph G with|V(G)|≥k+3 andκ(G)≥k+1 is(P≥3,k)-factor-critical covered if bind(G)≥4+k/3.
基金Supported by National Natural Science Foundation of China(Grant No.11371009)
文摘Let a,b,k,r be nonnegative integers with 1 ≤ a ≤b and r ≥ 2. Let G be a graph of order n with n 〉 （a＋b）（r（a＋b）-2）＋ak/a. In this paper, we first show a characterization for all fractional （a, b, k）-critical graphs. Then using the result, we prove that G is all fractional （a, b, k）-critical if δ（G） ≥ （r-1）b2/a ＋k and ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn＋ak/a＋b for any independent subset {xl, x2, .., xr} in G. Furthermore, it is shown that the lower bound on the condition ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn=ak/ a＋b is best possible in some sense, and it is an extension of Lu＇s previous result.
基金Supported by Natural Science Foundation of the Higher Education Institutions of Jiangsu Province(Grant No.10KJB110003)Jiangsu University of Science and Technology(Grant No.2010SL101J)+1 种基金National Natural Science Foundation of China(Grant No.71271119)National Social Science Foundation of China(Grant No.11BGL039)
文摘LetG be a graph,and k≥2 be a positive integer.A graph G is fractional independentset-deletable k-factor-critical（in short,fractional ID-k-factor-critical）,if G I has a fractional k-factor for every independent set I of G.The binding number bind（G）of a graph G is defined as bind（G）=min｜NG（X）｜｜X｜：=X V（G）,NG（X）=V（G）.In this paper,it is proved that a graph G is fractional ID-k-factor-critical if n≥6k 9 and bind（G）〉（3k 1）（n 1）kn 2k＋2.
