7

It is easy to obtain a site percolation model from a bond percolation model on a graph $G$ using the covering graph $G_c$ of $G$. I wondered if one can obtain any site percolation model from any site bond and I read in the Geoffrey Grimmett's book that is not true. Nevertheless he does not give any counterexample, and I cannot imagine someone. Can anybody give me a counter example?

José Milán
  • 71

1 Answers1

2

It seems as though "covering graph" is just another name for "line graph". As you can find in the "characterization" section of that wiki page, not every graph appears as a line graph of some other graph. A small 5 vertex example is shown there. I am not familiar with percolation, but I think this is basically what you are asking.

heynongman
  • 151
  • 1
    I understand the 5 vertex example, but I don't know if it works for infinite graphs (because in percolation theory usualy one works on infinite graphs). I mean, if I create a lattice using this 5 vertex graph infinite times, this lattice will be graph that is not a line graph? I think the resulting lattice is conformed by equilateral triangles and hexagons, a sublattice of the triangular lattice obtained erasing not-adjacent vertex and its respective edges. – José Milán Dec 14 '15 at 01:15