Note on 3-paths in plane graphs of girth 4

An (i,j,k)-path is a path on three vertices u, v and w in this order with deg(u)≤i, deg(v)≤j, and deg(w)≤k. In this paper, we prove that every connected plane graph of girth 4 and minimum degree at least 2 has at least one of the following: a (2,∞,2)-path, a (2,7,3)-path, a (3,5,3)-path, a (4,2,5)-path, or a (4,3,4)-path. Moreover, no parameter of this description can be improved. Our result supplements recent results concerning the existence of specific 3-paths in plane graphs.