Combining forbidden subgraphs with degree restrictions and neighborhood unionrestrictions,respectively,we prove the following results:（1） Let G be a 2-connected graph of order n,and 3≤c≤n.If for each induced subgr...Combining forbidden subgraphs with degree restrictions and neighborhood unionrestrictions,respectively,we prove the following results:（1） Let G be a 2-connected graph of order n,and 3≤c≤n.If for each induced subgraphL of order four of G（?）|V<sub>1</sub>（L）∩S<sub>c</sub>|≥2 if L≌K<sub>1,3</sub>,and |V（L）∩S<sub>c</sub>|≥1 if L≌P<sub>4</sub>,then thecircumference of G is at least c,where V<sub>1</sub>（L）is the set of vertices with degree 1 of L,S<sub>c</sub> isthe set of vertices with degree at least c/2 of G and P<sub>4</sub> is a path of order 4.（2） Let G be a 2-connected graph of order n,and n≥s+2.If for each induced subgraphL of G isomorphic to K<sub>1,3</sub>or P<sub>4</sub>,d<sub>L</sub>（u,v）=2（?）|N（u）∪N（v）|≥s,then the circumferencec （G） of G is at least s+2.Moreover,if n≥s+3 and s is odd,then c（G）≥s+3.展开更多
Let G be a graph of order n and X V(G). G is called X-cyclable if G has an X-cycle, i.e., a cycle containing all vertices of X. Define the parameters a(X) = max{|S| S is an independent vertex set in G[X] induced by X...Let G be a graph of order n and X V(G). G is called X-cyclable if G has an X-cycle, i.e., a cycle containing all vertices of X. Define the parameters a(X) = max{|S| S is an independent vertex set in G[X] induced by X}, σk(X) = min{∑ki=1dG(x.i| {x1, x1…, xk} is an independent vertex set in G[X]} and NCk(X) = min{|∪ki=1 NG(xi)| | {x1, x2…,xk} is an independent vertex set in G[X] }. Our main result is as follows: If G is a 1-tough graph and X V(G) with σ3(X)≥ n, then for every integer t ≥ 1, G has a cycle C containing at least min{|X|, (2|X| - n + 3δ + 1 - t), |X| + NCt(X) - a(X)} venices of X, where δ(X) = [σ3(X)]. This result further extends previous results in H.J. Broersma et al. in terms of X-cyclability. We also obtain that if G is a 1-tough graph with σ3 (X) ≥ n, then for every integer t ≥ 1, G has a cycle containing at least min{|X|, (4|X|- 2n+4δ(X) + 1 - 2t), NCt (X) +NCt (X)} vertices of X, where NCt (X) = min{|N(I) ∩X|| I is an independent set of t vertices of X}. Analogous results are established for 2-connected graphs.展开更多
基金A work supported by National Natural Science Foundation of China
文摘Combining forbidden subgraphs with degree restrictions and neighborhood unionrestrictions,respectively,we prove the following results:（1） Let G be a 2-connected graph of order n,and 3≤c≤n.If for each induced subgraphL of order four of G（?）|V<sub>1</sub>（L）∩S<sub>c</sub>|≥2 if L≌K<sub>1,3</sub>,and |V（L）∩S<sub>c</sub>|≥1 if L≌P<sub>4</sub>,then thecircumference of G is at least c,where V<sub>1</sub>（L）is the set of vertices with degree 1 of L,S<sub>c</sub> isthe set of vertices with degree at least c/2 of G and P<sub>4</sub> is a path of order 4.（2） Let G be a 2-connected graph of order n,and n≥s+2.If for each induced subgraphL of G isomorphic to K<sub>1,3</sub>or P<sub>4</sub>,d<sub>L</sub>（u,v）=2（?）|N（u）∪N（v）|≥s,then the circumferencec （G） of G is at least s+2.Moreover,if n≥s+3 and s is odd,then c（G）≥s+3.
基金the Natural Science Foundation of Yunnan Province.
文摘Let G be a graph of order n and X V(G). G is called X-cyclable if G has an X-cycle, i.e., a cycle containing all vertices of X. Define the parameters a(X) = max{|S| S is an independent vertex set in G[X] induced by X}, σk(X) = min{∑ki=1dG(x.i| {x1, x1…, xk} is an independent vertex set in G[X]} and NCk(X) = min{|∪ki=1 NG(xi)| | {x1, x2…,xk} is an independent vertex set in G[X] }. Our main result is as follows: If G is a 1-tough graph and X V(G) with σ3(X)≥ n, then for every integer t ≥ 1, G has a cycle C containing at least min{|X|, (2|X| - n + 3δ + 1 - t), |X| + NCt(X) - a(X)} venices of X, where δ(X) = [σ3(X)]. This result further extends previous results in H.J. Broersma et al. in terms of X-cyclability. We also obtain that if G is a 1-tough graph with σ3 (X) ≥ n, then for every integer t ≥ 1, G has a cycle containing at least min{|X|, (4|X|- 2n+4δ(X) + 1 - 2t), NCt (X) +NCt (X)} vertices of X, where NCt (X) = min{|N(I) ∩X|| I is an independent set of t vertices of X}. Analogous results are established for 2-connected graphs.
