## Vertex Coloring of graph fractional powers ★★★

Conjecture   Let be a graph and be a positive integer. The power of , denoted by , is defined on the vertex set , by connecting any two distinct vertices and with distance at most . In other words, . Also subdivision of , denoted by , is constructed by replacing each edge of with a path of length . Note that for , we have .
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.

## Partial List Coloring ★★★

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 .
Conjecture    Let be a graph with list chromatic number and . Then  