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 p_{\mathrm{c}}(G_{*}) < p_{\mathrm{c}}(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 p_{\mathrm{u}}(G) + p_{\mathrm{c}}(G) \ge 1 , with equality if and only if the graph G_{\Delta} , obtained from G by emptying all separating triangles, is a triangulation. Here, p_{\mathrm{u}} is the critical probability for the existence of a unique infinite open cluster.