Herscovici Conjecture on Pebbling

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 $\pi_t(G)$(least positive integer) pebbles, if we can transfer $t$ pebbles to any target through a number of pebbling moves then $\pi_t(G)$ 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.