Graham’s pebbling conjecture on product of thorn graphs of complete graphs

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 pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p 1 , p 2 , … , p n be positive integers and G be such a graph, V ( G ) = n . The thorn graph of the graph G , with parameters p 1 , p 2 , … , p n , is obtained by attaching p i new vertices of degree 1 to the vertex u i of the graph G , i = 1 , 2 , … , n . 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 for a thorn graph of the complete graph with every p i > 1 ( i = 1 , 2 , … , n ) by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every p i > 1 ( i = 1 , 2 , … , n ) .