#### Title

Graham’s Conjecture, Lemke Graphs, and Algorithms

#### Faculty Mentor(s)

Dr. Charles Cusack and Dr. Airat Bekmetjev

#### Document Type

Poster

#### Event Date

4-15-2016

#### Abstract

Graph pebbling is a game on a connected graph *G* in which pebbles are placed on the vertices of *G*. A *pebbling move* consists of removing two pebbles from any vertex and adding one to an adjacent vertex. A configuration of pebbles is *r-solvable* if for a given target vertex *r*, there is a sequence of pebbling moves so that at least one pebble can be placed on *r*. The *pebbling number* of a graph *G* is the smallest integer π(*G*) such that any configuration that uses π(*G*) pebbles is *r-solvable* for any *r * * V *(*G*)*.* A long standing conjecture in graph pebbling is Graham’s Conjecture. It states that given any two graphs *G *and *H*, π(*G*¨*H*)* *≤ π(*G*)π(*H*)*,* where *G*¨*H* is the Cartesian product of graphs. A graph *G* satisfies the *two-pebbling property* if two pebbles can be placed on any vertex *v * *V *(*G*) given any configuration of 2π(*G*)−*q*+1 pebbles, where *q* is the number of vertices that have at least one pebble. The only known graphs that do not satisfy the two-pebbling property are called Lemke graphs (*L _{k}*). We will show that Graham’s conjecture holds for such families as

*L*¨

_{k}*K*and several others.

_{n}#### Recommended Citation

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## Comments

This research was supported by the Howard Hughes Medical Institute through the Undergraduate Science Education Program.