Pebbling a cartesian product

Importance: High ✭✭✭
Author(s): Graham, Ronald L.
Subject: Graph Theory
Keywords: pebbling
zero sum
Recomm. for undergrads: no
Posted by: mdevos
on: September 24th, 2007

We let $p(G)$ denote the pebbling number of a graph $G$.

\begin{conjecture} $p(G_1 \Box G_2) \le p(G_1) p(G_2)$. \end{conjecture}

The \emph{pebbling number} of a graph $G$, denoted $p(G)$, is the smallest integer $k$ so that however $k$ pebbles are distributed onto the vertices of $G$, it is possible to move a pebble to any vertex by a sequence of steps, where in each step we remove two pebbles from one vertex, and then place one on an adjacent vertex. The \emph{cartesian product} of two graphs $G_1$ and $G_2$, denoted $G_1 \Box G_2$, is the graph with vertex set $V(G_1) \times V(G_2)$ and an edge from $(v,w)$ to $(v',w')$ if either $v=v'$ and $w \sim w'$ (in $G_2$) or $w=w'$ and $v \sim v'$ (in $G_1$).

Graph Pebbling arose out of the study of zero-sum subsequences in groups, but has proved to be a rich and interesting topic in graph theory. See Glenn Hurlbert's wonderful \href[graph pebbling page]{http://mingus.la.asu.edu/~hurlbert/pebbling/pebb.html} for much more on this topic (and this problem in particular). Graham's conjecture was stated in one of the first papers on this subject by Fan Chung [C], and has since generated considerable interest.

There are a number of partial results toward this conjecture. Chung [C] proved that $p(P_{d_1+1} \Box P_{d_2+1} \ldots \Box P_{d_{\ell}+1}) = 2^{d_1 + d_2 \ldots + d_{\ell}}$, thus settling the conjecture for products of paths (here $P_n$ denotes a path with $n$ vertices). It is also known when $G_1,G_2$ are both trees, both cycles, and for graphs with high minimum degree. Again, we encourage the interested reader to see \href[the graph pebbling page]{http://mingus.la.asu.edu/~hurlbert/pebbling/pebb.html} for more details.

Bibliography

*[C] F. Chung, \href[Pebbling in hypercubes]{http://mingus.la.asu.edu/~hurlbert/pebbling/papers/Chun_PIH.pdf} SIAM J. Disc. Math. 2 (1989), 467--472.


* indicates original appearance(s) of problem.