Pebbling on Cartesian Product
The graph G, as is shown below on the left, has 2-connectivity and diameter 2. It is verified that this graph is class-1. However, whether the graph product of G with itself is a class-0 graph remains an interesting problem.
If we observe the product graph closely, we can see that the graph is composed of 14 Cartesian product graphs of K3 by K3. Notice that the graph product of K3 by K3 is class-0 graph.
To verify that the product graph holds for Graham's conjecture, we need to show that any distribution of 49 pebbles over the vertices of the product graph is solvable. The approach we use was proof my contradiction,and the structure of the K3 by K3 graph offers a very useful way of counting. The details are presented in my second presentation. Click here.
2k-Cycles in a Bipartite Graph
The problem is stated in my first presentation. Click here.