標題: "Pancyclicity, Panconnectivity, and Panpositionability for General Graphs and Bipartite Graphs" 作者: Kao, Shin-ShinLin, Cheng-KuanHuang, Hua-MinHsu, Lih-Hsing資訊工程學系Department of Computer Science 公開日期: 1-七月-2012 摘要: "A graph G is pancyclic if it contains a cycle of every length from 3 to vertical bar V(G)vertical bar inclusive. A graph G is panconnected if there exists a path of length l joining any two different vertices x and y with d(G)(x,y) <= l <= vertical bar V(G)vertical bar - 1, where d(G)(x,y) denotes the distance between x and y in G. A hamiltonian graph G is panpositionable if for any two different vertices x and y of G and any integer k with d(G)(x,y) <= k <= vertical bar V(G)vertical bar/2, there exists a hamiltonian cycle C of G with d(C)(x,y) = k, where d(C)(x,y) denotes the distance between x and y in a hamiltonian cycle C of G. It is obvious that panconnected graphs are pancyclic, and panpositionable graphs are pancyclic. The above properties can be studied in bipartite graphs after some modification. A graph H = (V-0 boolean OR V-1, E) is bipartite if V(H) = V-0 boolean OR V-1 and E(H) is a subset of {(u, v) vertical bar u is an element of V-0, v is an element of V-1}. A graph is bipancyclic if it contains a cycle of every even length from 4 to 2 . left perpendicular vertical bar V(H)vertical bar/2right perpendicular inclusive. A graph H is bipanconnected if there exists a path of length 1 joining any two different vertices x and y with d(H)(x,y) <= l <= vertical bar V(H)vertical bar - 1, where d(H)(x, y) denotes the distance between x and y in H and 1 dH(x,y) is even. A hamiltonian graph H is bipanpositionable if for any two different vertices x and y of H and for any integer k with d(H)(x,y) <= k < vertical bar V(H)vertical bar/2, there exists a hamiltonian cycle C of H with d(G)(x,y) = k, where d(G)(x,y) denotes the distance between x and y in a hamiltonian cycle C of H and k - d(H)(x,y) is even. It can be shown that bipanconnected graphs are bipancyclic, and bipanpositionable graphs are bipancyclic. In this paper, we present some examples of pancyclic graphs that are neither panconnected nor panpositionable, some examples of panconnected graphs that are not panpositionable, and some examples of graphs that are panconnected and panpositionable, for nonbipartite graphs. Corresponding examples for bipartite graphs are discussed. The existence of panpositionable (or bipanpositionable, resp.) graphs that are not panconnected (or bipanconnected, resp.) is still an open problem." URI: http://hdl.handle.net/11536/16755 ISSN: 0381-7032 期刊: ARS COMBINATORIA Volume: 105 Issue: 起始頁: 結束頁: 231 顯示於類別： 期刊論文