The Problems of Rankings on Sierpinski graphs

• Main Authors: YO-Lin Lin, 林佑霖
• Other Authors: Justie Su-Tzu Juan
• Format: Others
• Language: en_US
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/17491090115707804237

碩士 === 國立暨南國際大學 === 資訊工程學系 === 98 === For a graph G = (V;E), a vertex (edge, respectively) t-ranking is a coloring c : V ! f1; 2; : : : ; tg (c0 : E ! f1; 2; : : : ; tg, respectively) such that, for any two vertices u and v (edges ex and ey, respectively) with c(u) = c(v) (c0(ex) = c0(ey), respectively), every path between them contains an intermediate vertex w (edge ew, respectively) with c(w) > c(u) (c0(ew) > c0(ex), respectively). The vertex ranking number r(G) (edge ranking number 0 r(G), respectively) is the smallest value of t such that G has a vertex (edge, respectively) t-ranking. The problem to nd r(G) (0 r(G), respectively) for a graph G is called the vertex ranking problem (the edge ranking problem, respectively). A partition of V is a set of nonempty subsets of V such that every vertex in V is in exactly one of these subsets. In this thesis, we introduce two relations for ranking number of a graph. One is between vertex ranking number and vertex partitions, the other is between edge ranking number and vertex partitions. By using the proposed recurrence formulas, we derived two results. One is the edge ranking number of the Sierpinski graph 0 r(S(n; k)) = n0 r(Kk) for any n; k > 2, and the other is the bound of vertex ranking number of Sierpinski graphs (n2) r(Kk)+r(S00(n; k)) 6 r(S(n; k)) 6 (n2)0 r(Kk)+r(S00(n; k))+1 for any n > 2 and k > 3 where S00(n; k) is the subgraph of S(n; k) by removing all vertices of degree k 1 from S(n; k).