Projective-Planar Graphs with No K3, 4-Minor

There are several graphs H for which the precise structure of graphs that do not contain a minor isomorphic to H is known. In particular, such structure theorems are known for K5 [13], V8 [8] and [7], the cube [3], the octahedron [4], and several others. Such characterizations can often be very useful, e.g., Hadwiger’s conjecture for k = 4 is verified by using the structure for K5-free graphs, and the structure theorem for V8-free graphs is used to characterize how projective-planar graphs may be re-embedded in the projective plane [5]. Characterizations of K6-free graphs and Petersen-free graphs are highly sought-after results, mostly due to their connections with Hadwiger’s conjecture and Tutte’s 4-flow conjecture. Such characterizations seem to be very difficult. The Petersen graph and K6 belong to a collection of seven graphs known as the Petersen Family of graphs (see [9]). They are all graphs obtained by sequences of Y∆ and ∆Y operations on the Petersen graph. The difficulty of characterizing K3,4-free graphs seems to lie between the characterizations of H-free graphs mentioned in the first paragraph and H-free graphs for H in the Petersen Family. In this paper we give an exact structure for projective-planar graphs that are K3,4-free (Theorem 3.4 along with Propositions 3.1 and 3.2). The authors hope that this might be a first step in a complete structure theorem of K3,4-free graphs. The non-projective-planar K3,4-free graphs might be characterized using the known list of 35 minor-minimal non-projective planar graphs in [1] and [2]. Another possible point of interest for characterizing K3,4-free graphs might be the following. A k-separation (B1, B2) in a graph G is called flat if the subgraph of G induced by some Bi along with a vertex of degree k attached to the k vertices of V (B1) ∩ V (B2) is a planar graph. In Section 3 we will see that a 3-connected graph G is K3,4-free iff every 3-separation in every 3-connected minor of G is flat. ∗Dept. of Mathematics, The Ohio State University, Columbus OH 43210, [email protected] †Dept. of Mathematics and Statistics, Wright State University, Dayton OH 45435, [email protected]