Non-self-touching paths in plane graphs

A path in a graph G is called non-self-touching if two vertices are neighbours in the path if and only if they are neighbours in the graph. We investigate the existence of doubly infinite non-self-touching paths in infinite plane graphs.The matching graph G* of an infinite plane graph G is obtained by adding all diagonals to all faces, and it plays an important role in the theory of site percolation on G . The main result of this paper is a necessary and sufficient condition on G for the existence of a doubly infinite non-self-touching path in G* that traverses some diagonal. This is a key step in proving, for quasi-transitive G , that the critical points of site percolation on G and G* satisfy the strict inequality pc(G*) < pc(G) , and it complements the earlier result of Grimmett and Li [Random Struct. Alg. 65 (2024) 832–856], proved by different methods, concerning the case of transitive graphs. Furthermore, it implies, for quasi-transitive graphs, that pᵤ(G) + pc(G) ≥ 1 , with equality if and only if the graph GΔ , obtained from G by emptying all separating triangles, is a triangulation. Here, pᵤ is the critical probability for the existence of a unique infinite open cluster.