Now we can define the fractional power of a graph as follows:
Let be a graph and . The graph is defined by the power of the subdivision of . In other words .
Conjecture. Let be a connected graph with and be a positive integer greater than 1. Then for any positive integer , we have .
In , it was shown that this conjecture is true in some special cases.
Let be a simple graph, and for every list assignment let be the maximum number of vertices of which are colorable with respect to . Define , where the minimum is taken over all list assignments with for all .