Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f（G） is the smallest number m such that for every distribution of m pebbles and every vertex v, a pebble can be moved to v. A graph G is said to have the 2-pebbling property if for any distribution with more than 2f（G） - q pebbles, where q is the number of vertices with at least one pebble, it is possible, using pebbling moves, to get two pebbles to any vertex. Snevily conjectured that G（s, t） has the 2- pebbling property, where G（s, t） is a bipartite graph with partite sets of size s and t （s 〉 t）. Similarly, the ~,pebbling number fl（G） is the smallest number m such that for every distribution of m pebbles and every vertex v, ~ pebbles can be moved to v. Herscovici et al. conjectured that fl（G） ≤ 1.5n ＋ 8l -- 6 for the graph G with diameter 3, where n = IV（G）I. In this paper, we prove that if s ≥ 15 and G（s,t）
Let n 〉 r, let lr --- （dl,d2,-,dn） be a non-increasing sequence of nonnegative integers and let Kr＋l - e be the graph obtained from Kr＋l by deleting one edge. If zr has a realization G containing Kr＋l - e as a subgraph, then r is said to be potentially Kr＋l - e-graphic. In this paper, we give a characterization for a sequence π to be potentially Kr＋l - e-graphic.
Let r 3, n r and π = (d1, d2, . . . , dn) be a graphic sequence. If there exists a simple graph G on n vertices having degree sequence π such that G contains Cr (a cycle of length r) as a subgraph, then π is said to be potentially Cr-graphic. Li and Yin (2004) posed the following problem: characterize π = (d1, d2, . . . , dn) such that π is potentially Cr-graphic for r 3 and n r. Rao and Rao (1972) and Kundu (1973) answered this problem for the case of n = r. In this paper, this problem is solved completely.
YIN JianHua Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou 570228, China
