In this figure, it finds a vertex to another vertex k shortest path algorithm. Provided there are n vertices and edges in the diagram. If the path loops, the time complexity of the algorithm is allowed O（w ＋ n log 2...In this figure, it finds a vertex to another vertex k shortest path algorithm. Provided there are n vertices and edges in the diagram. If the path loops, the time complexity of the algorithm is allowed O（w ＋ n log 2 n ＋ kw log 2 k）. If the request path does not contain the loop, the time complexity of the algorithm O（kn（w ＋ n log2 n）＋ kw log2 k）. The algorithm utilizes a simple extension of the Dijkstra algorithm determined the end of the length of the shortest path to the other vertices, and then, based on these data, branch and bound method to identify the required path. Experimental results show that the actual running time has relations with the structure of FIG.展开更多
An r-circular coloring of a graph G is a map f from V(G) to the set of open unit intervals of an Euclidean circle of length r, such that f(u) ∩ f(v) = Ф whenever uv ∈ E(G). Circular perfect graphs are defined analo...An r-circular coloring of a graph G is a map f from V(G) to the set of open unit intervals of an Euclidean circle of length r, such that f(u) ∩ f(v) = Ф whenever uv ∈ E(G). Circular perfect graphs are defined analogously to perfect graphs by means of two parameters, the circular chromatic number and the circular clique number. In this paper, we study the properties of circular perfect graphs. We give (1) a necessary condition for a graph to be circular perfect, (2) some circular critical imperfect graphs, and (3) a characterization of graphs with the property that each of their induced subgraphs has circular clique number the same as its clique number, and then the two conjectures that are equivalent to the perfect graph conjecture.展开更多
An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define ...An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k. The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.展开更多
Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, a free field representation of toroidal Lie superalgebras of type A(m, n) is constructed using both vertex operators and bosonic fields.
文摘In this figure, it finds a vertex to another vertex k shortest path algorithm. Provided there are n vertices and edges in the diagram. If the path loops, the time complexity of the algorithm is allowed O（w ＋ n log 2 n ＋ kw log 2 k）. If the request path does not contain the loop, the time complexity of the algorithm O（kn（w ＋ n log2 n）＋ kw log2 k）. The algorithm utilizes a simple extension of the Dijkstra algorithm determined the end of the length of the shortest path to the other vertices, and then, based on these data, branch and bound method to identify the required path. Experimental results show that the actual running time has relations with the structure of FIG.
基金This research is supported partially by National Natural Science Funds of China(10001035 and 10371055).
文摘An r-circular coloring of a graph G is a map f from V(G) to the set of open unit intervals of an Euclidean circle of length r, such that f(u) ∩ f(v) = Ф whenever uv ∈ E(G). Circular perfect graphs are defined analogously to perfect graphs by means of two parameters, the circular chromatic number and the circular clique number. In this paper, we study the properties of circular perfect graphs. We give (1) a necessary condition for a graph to be circular perfect, (2) some circular critical imperfect graphs, and (3) a characterization of graphs with the property that each of their induced subgraphs has circular clique number the same as its clique number, and then the two conjectures that are equivalent to the perfect graph conjecture.
基金Foundation item： the National Natural Science Foundation of China （No. 10626029）.
文摘An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k. The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.
基金supported by the National Natural Science Foundation of China(No.11271138,No.11301393)the Simons Foundation(No.198219)the domestic visiting scholar professional development project of colleges and Universities in Zhejiang Province(No.FX2014099)
文摘Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, a free field representation of toroidal Lie superalgebras of type A(m, n) is constructed using both vertex operators and bosonic fields.
