Strong hub cover pebbling number
Runze Wang
TL;DR
The paper introduces the strong hub set and its associated strong hub cover pebbling number $h_s^*(G)$, requiring a path between any two vertices whose internal vertices all lie in the hub. It derives exact values for common graph families, namely $h_s^*(P_n)=0$ for $n\le 2$ and $h_s^*(P_n)=2^{n-1}-1$ for $n\ge 3$, $h_s^*(S_n)=n+1$, and $h_s^*(B_n)=2n+3$, using inductive and constructive pebbling arguments. It also provides a conjecture for cycles, proposing explicit formulas for $h_s^*(C_n)$ depending on the parity of $n$. By strengthening hub-based connectivity, the work yields precise pebbling thresholds for paths, stars, and books and motivates further study on cycles within the graph pebbling framework.
Abstract
In a graph $G$, we define a set of vertices to be a \emph{strong hub set} if for any two vertices in $G$, we can find a path between them whose internal vertices are all in this set. We define the \emph{strong hub cover pebbling number} of $G$, denoted by $h_s^*(G)$, to be the smallest $t$ such that for any initial configuration with $t$ pebbles on $G$, we can make some pebbling moves (a pebbling move consists of removing two pebbles from a vertex $v$ and adding one pebble to another vertex adjacent to $v$) so that there is a strong hub set with every vertex in it having a pebble. We determine the strong hub cover pebbling numbers of paths, stars, and books.