Zum Basisproblem der nicht in die projektive Ebene einbettbaren Graphen II

Abstract

Let Γ be the set of finite graphs which are not embeddable into the projective plane. Let M( Γ >) denote the set of the minimal graphs of Γ. In former papers [3] and [5] K. Wagner has proved the following result: Each minimal graph G of M( Γ >) which is different from the graphs G 1, G 2, …, G 12, G 14, D, F fulfils the three following conditions: 1. (1) G is at least 3-vertex connected. 2. (2) In G exists one vertex v, for which the graph G v obtained from G by deleting v and all edges of G which are incident with v contains a U( K 3,3). 3. (3) For each vertex v of G, the graph G v is either planar or contains a U( K 3,3). In the following we investigate the character of these graphs G. For this some essential terms are necessary. Let G be a graph with the vertex set V and G′ a subgraph of G with the vertex set V′. Then G′ is called a subgraph of the nth class (designated by G′ ⊆ ( n) G ) if and only if the chromatic number of the subgraph G( V − V′) of G spanned by V − V′ is equal to n. Let X be a connected component of the spanned subgraph G( V − V′) of G and let Y be the bipartite graph consisting of all those edges of G which combine X with G′. Then the graph Q = X ⌣ Y is called a relative component of G in relation to G′ (or “Querstück” of G in relation to G′). Let z 1, z 2, z 3, z 4, a, b be six different vertices of G from which z 1, z 2, z 3, z 4 lie on a circuit Z of G. Let W 1, W 2, W 3 denote three paths of G with W 1 = W( z 1, z 2), W 2 = W( z 3, z 4), and W 3 = W( a, b) 1 1 W( z 1, z 2) means that z 1 and z 2 are the endpoints of this path. Similar: W( z 3, z 4) and W( a, b). . The subgraph W 1 ⌣ W 2 ⌣ W 3 of G is called a “Kreuzhaube” H with the basis points z 1, z 2, z 3, z 4 if and only if 1. (1) W 1 ⌣ Z = {z 1, z 2} , 2. (2) W 2 ⌣ Z = {z 3, z 4} , 3. (3) W 1 ⌣ W 2 = W 3 ⌣ Z = ⊘ , 4. (4) W 3 ⌣ W 1 = {a} , 5. (5) W 3 ⌣ W 2 = {b} , 6. (6) The two pairs of basis points z 1, z 2 and z 3, z 4 are separated from each other on Z (Fig. 1). ▪ Having characterized minimal graphs of M( Γ >) in “Satz 1”, we prove the following theorem: Let G be a minimal graph of M(Γ >) which is different from G 1,G 2,…,G 12 and G 14 and which contains a G′ = U(K 3,3) as a subgraph of the nth class with n ≥ 1. Then only one of the following three statements holds for each of the subgraphs G′ = U(K 3,3) and for each relative component Q of G in relation to G′ : 1. (1) Q is a star (“Stern”) . 2. (2) Q contains a “Kreuzhaube” H which has only its four basis-points with G′ = U(K 3,3) in common. To be more precise: If all those vertices of Q which do not lie on G′ are contracted in one single vertex, the result is that Q becomes a star Q ∗ and G becomes G ∗ (with fewer vertices than G and therefore G ∗ being embeddable into the projective plane), then Q in relation with each embedding of G ∗ into the projective plane contains a “Kreuzhaube” which has only its four basis-points in common with that circuit of the G′ on which the endpoints of Q ∗ lie at the embedding of G ∗ . 3. (3) G = P. In this equation P means the graph of M(Γ >) which one obtains from two disjoint K 3,3 by identifying two neighboring edges of the one K 3,3 with two neighboring edges of the other K 3,3.