Herscovici Conjecture on Pebbling
I. Dhivviyanandam, A. Lourdusamy, S. Kither Iammal, K. Christy Rani
TL;DR
The paper addresses Herscovici's pebbling conjecture in the setting of total graphs, focusing on the total graph of a path $T(P_n)$. It derives exact formulas for the pebbling numbers, showing $π(T(P_n))=2^{n-1}+(n-2)$ and $π_t(T(P_n))=t2^{n-1}+(n-2)$, and proves that $T(P_n)$ satisfies both the $2$-pebbling and the $2t$-pebbling properties. Extending to products, it proves the conjectured inequality $π_{st}(T(P_n)\times T(P_m))\le π_s(T(P_n))\,π_t(T(P_m))$, thereby validating Herscovici's conjecture for this graph family. These results advance pebbling theory on total graphs and provide a concrete, analyzable case supporting the conjecture in a natural graph-product setting.
Abstract
Consider a configuration of pebbles on the vertices of a connected graph. A pebbling move is to remove two pebbles from a vertex and to place one pebble at the neighbouring vertex of the vertex from which the pebbles are removed. For a positive integer $t$, with every configuration of $π_t(G)$(least positive integer) pebbles, if we can transfer $t$ pebbles to any target through a number of pebbling moves then $π_t(G)$ is called the $t$-pebbling number of $G$. We discuss the computation of the $t$-pebbling number, the $2t-$ pebbling property and Herscovici conjecture considering total graphs. \bigskip \noindent Keywords: pebbling moves, $t$- pebbling number, $2t$-pebbling property, Herscovici conjecture, total graphs.