Discrete Jordan Curve Theorems

There has been recent interest in combinatorial versions of classical theorems in topology. In particular, Stahl [S] and Little [3] have proved discrete versions of the Jordan Curve Theorem. The classical theorem states that a simple closed curve y separates the 2-sphere into two connected components of which y is their common boundary. The statements and proofs of the combinatorial versions in [3, 51 are given in terms of permutation pairs and colored graphs (see Sect. 4). In this paper short proofs of three graph theoretic versions of the Jordan Curve Theorem are given. A graph G may have multiple edges but no loops. It is understood that each vertex in a cycle has degree 2. A cycle y in a graph G will be said to have the First Jordan Curve Property (JCPl) if there exist connected proper subgraphs Z and 0 of G such that In 0 = y and Zu 0 = GO, where G, is the connected component of G containing y. In particular, any path from a vertex of Z to a vertex of 0 contains a vertex of y. A family C of