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Graham’s pebbling conjecture on product of thorn graphs of complete graphs
Authors:Zhiping Wang  Zhongtuo Wang
Institution:a Department of Mathematics, Dalian Maritime University, 116026, Dalian, PR China
b College of Traffic and Logistics Engineering, 116026, Dalian, PR China
c Department of Foundational Education, Yantai Nanshan University, 265713, Yantai, PR China
d School of Management, Dalian University of Technology, 116024, Dalian, PR China
Abstract: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 p1,p2,…,pn be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p1,p2,…,pn, is obtained by attaching pi new vertices of degree 1 to the vertex ui 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 View the MathML source 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 View the MathML source.
Keywords:Pebbling number  Graham&rsquo  s conjecture  Thorn graph  Complete graph  Cartesian product
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