The pebbling number of a graph G,f(G),is the least n such that,no matter how n pebbles are placed on the vertices of G,we can move a pebble to any vertex by a sequence of moves,each move taking two pebbles off one ver...The pebbling number of a graph G,f(G),is the least n such that,no matter how n pebbles are placed on the vertices of G,we can move a pebble to any vertex by a sequence of moves,each move taking two pebbles off one vertex and placing one on an adjacent vertex.Graham conjectured that for any connected graphs G and H,f(G×H)≤f(G)f(H).We show that Graham's conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property.As a corollary,Graham's conjecture holds when G and H are complete bipartite graphs.展开更多
基金This work was supported by the National Natural Science Foundation of China (Grant Nos. 49873002, 10001005).
文摘The pebbling number of a graph G,f(G),is the least n such that,no matter how n pebbles are placed on the vertices of G,we can move a pebble to any vertex by a sequence of moves,each move taking two pebbles off one vertex and placing one on an adjacent vertex.Graham conjectured that for any connected graphs G and H,f(G×H)≤f(G)f(H).We show that Graham's conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property.As a corollary,Graham's conjecture holds when G and H are complete bipartite graphs.
