Face Width and Graph Embeddings of face-width 2 and 3

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1 Face Width and Graph Embeddings of face-width 2 and 3 Instructor: Robin Thomas Scribe: Amanda Pascoe 3/12/07 and 3/14/07 1 Representativity Recall the following: Definition 2. Let Σ be a surface, G a graph, and ψ : G Σ an embedding (not necessarily 2-cell). The representativity or face-width of ψ is the maximum integer k such that every non-null-homotopic closed curve intersects ψ(g) at least k times. Example. The 5-grid embedded in P has high representativity; precisely, its face-width is 5. See Figure 1. A cut along the dotted line pictured shows that fw(φ) 5. We will consider later the reverse inequality. We have shown that if G is 2-connected and G Σ has face width at least 2 then every face is bounded by a cycle. This is a generalization of the fact that faces of 2-connected planar graphs are bounded by cycles. In the sphere, face width is infinity, so of course any 2-connected graph has satisfies fw(g) Cycle-Double-Covers Recall the Cycle-Double-Cover Conjecture: Conjecture. Every 2-connected graph has a list of cycles (not necessarily distinct) such that every edge belongs to exactly 2 members of the list. Try the following easy exercise: It is enough to show the conjecture for cubic graphs. In fact, it is even enough to show for cubic graphs that are not 3-edge-colorable ( snarks ). Fact. A cycle-double cover C in a cubic graph defines an embedding in a surface: the members of C are precisely the facial cycles. Proof. Consider a vertex v. Each edge incident with v is in exactly two cycles C 1, C 2. C 1 and C 2 can t both share another edge incident with v because then 1

2 the remaining edge could not be in any cycle. Hence the picture looks like Figure 2. Now, just glue a disk onto each cycle to create a surface. Remark. For a vertex v, define an auxiliary graph H v on N(v) by saying x y if some cycle in C includes a path x... v... y. Then H v is 2-regular. A cycledouble cover C is the collection of face boundaries in some embedding G Σ if and only if for every vertex v, H v is connected (H v is a single cycle, and not a collection of cycles). There are several generalizations of the CDC (Cycle-Double-Cover) Conjecture, with varying degrees of ambition. Conjecture (Strong Embedding Conjecture). Every 2 connected graph G has a 2-cell ( strong ) embedding with face width at least 2 in some surface Σ. Notice that if this conjecture holds then the face boundaries form a CDC. The assumption that face width is at least 2 ensures that the cover is made up of cycles. Conjecture (Strong Orientable Embedding Conjecture). The same as above, except with the additional requirement that Σ is orientable. Conjecture. Same as above, except the embedding is 5-face-colorable. More precisely, every 2 connected graph G has a strong 5-face-colorable embedding with face width at least 2 in some surface Σ (can require either Σ orientable or not). 2.2 Separating Cycles The following conjecture is known for Σ = N 2, the Klein bottle, but is open for all other surfaces with nontrivial separating curves. Conjecture (Separating Cycle Conjecture). Let G Σ have face width at least 3 and let Σ have a nontrivial separating curve (this is independent of the embedding). Then G has a nontrivial separating cycle. Note that requiring Σ to have a nontrivial separating curve eliminates only S 0, S 1, N 1, since these are the only surfaces that don t have such a curve. Note also that it is enough to show that the conjecture holds for 2 cell embeddings, since for an arbitrary embedding we can find a separating cycle from the separating cycle of a corresponding 2-cell embedding ψ using the rotation scheme for ψ. Minimal embeddings with face width at least k for every k in S 1 were described by A. Schrijver. He relates the embeddings to the geometry of numbers. 2

3 Some terminology: 1. k-representativity is equivalent to fw k. 2. If G Σ S 0 and a walk W bounds a disk, let int(w ) denote the closure of this disk. Clearly, if W int(w ) then int(w ) int(w ), since that disk is unique for Σ S 0. Theorem. Let G Σ be 2-representative (and therefore 2-cell), and Σ S 0. Then every facial walk W includes a contractible (null-homotopic) cycle C such that W int(c). Proof. Let the walk W bound a face f. We will in fact show by induction that W, f int(c). If every vertex of W is distinct then W is precisely the cycle that satisfies the theorem. Hence WMA there exists a vertex x W that appears twice. We can find a closed curve ψ inside f from x to x. See Figure 3 and Figure 4 for an example of what this might look like. The curve ψ meets G exactly once. Since G is 2-representative, this implies that ψ is null-homotopic, and thus bounds a disk. This has the important consequence that no vertex y x lies both between these two appearances of x and after the second appearance of x. More precisely, if x, y, x, w appear in this order on W then y w. Therefore we can choose a subwalk W = {v 1, v 2,..., v k } of W such that v 1 = v k and W belongs to the disk bounded by the curve ψ constructed above. See again Figure 3. Choose W subject to that k is minimum. The choice of k ensures that v i v j for 1 i < j k 1. In other words, W is a cycle. Remove W and proceed by induction. That is, delete v 2, v 3,..., v k 1 from G (and W ). By induction the new facial walk, say W 1, includes the cycle C with W 1 int(c) and f 1 int(c), where f 1 is the face bounded by W 1. Now C is the cycle we want, since W f 1 int(c), and so we are done. Corollary. If G Σ has representativity at least 2 and Σ S 0 then G has a unique block B such that B contains a noncontractible cycle. The induced embedding of B is 2-representative and each block B B of G is a plane subgraph contained in int(c) for some facial cycle C of B. Moreover, fw(b) = fw(g). Proof. Take the union of the cycles from the previous proposition. Each facial walk gives a unique cycle C; let B be the union of these cycles. If B is either not 2 connected or not 2 representative then there exists a closed curve ψ meeting B at most once such that either ψ is non-null-homotopic (which would contradict the fact that G is 2 representative) or ψ separates B (which contradicts the definition of B). See Figure 6. Note that there may be some planar pieces of G living in the faces of B. Take a curve attaining fw(b). If it intersects these planar pieces of G then it can be redrawn to avoid G \ B. See Figure 7. 3

4 2.3 Connectivity and embeddings Reminder: Let ψ : G Σ. Then fw(ψ) 2 and G is 2-connected every face is a disk bounded by a cycle. This is called a closed 2-cell embedding. Theorem. Let G Σ. Then fw 2 and G is 3 connected every face is a disk bounded by a cycle and every 2 facial cycles are disjoint or meet in a vertex or meet in an edge. The proof is analagous to the 2-representative version. Proof (Sketch). ( ) Suppose two facial cycles C 1, C 2 bounding faces f 1, f 2, respectively, intersect in a bad way. See Figure 8. Clearly, two distinct facial cycles can t intersect in a triangle, so there must be a pair of vertices x, y C 1 C 2 such that the edge xy / C 1 C 2. Find a closed curve through x, y that goes through f 1 and f 2. This curve intersects G in 2 vertices, and thus by 3 representativity bounds a disk. By 3 connectivity the disk contains (no vertices and) exactly one edge of G, which implies that C 1 C 2 is an edge, a contradiction. ( ) Similar. If fw 2 then there exists a non-null-homotopic closed curve ψ meeting G at most twice. Then G divides ψ into at most two pieces and these pieces belong to faces whose boundaries intersect in an illegal way. See Figure 9. If G is not 3-connected then there exists a closed curve through the cut set of G (separating G and intersecting it twice - see Figure 10). Proceed as above. Definition 3. If G is simple and 3 connected then any 3-representative embedding G Σ is called polyhedral. Theorem (Steinitz). A graph G is the 1-skeleton (vertices and edges) of a convex polyhedron in R 3 if and only if G is 3-connected and planar. Theorem. If G Σ is polyhedral then all facial cycles are peripheral (induced and non-separating). Example. The highlighted cycle is peripheral but not facial. Figure 11. Proof. Let C be a facial cycle. Then if C is not induced, there exists an edge e E(G) \ E(C) with both ends in C. See Figure 12. Let f be incident with e. The boundaries of f, f meet in an illegal way. A similar argument shows that C is nonseparating. A consequence of this theorem is that a planar graph does not have a polyhedral embedding in a nonplanar surface. Corollary. If G Σ is polyhedral and G is planar then Σ = S 0. 4

5 Proof. Let G S 0 be the unique planar drawing of G (it is unique by the 3-connectivity of G). The number of faces is maximized when the genus of the embedded surface is minimal by Euler s formula ( V + F = E + 2 2g). Hence F (G) F (G ). But by the previous theorem, every face of G is a face of G and so Σ = S 0. This implies another way to think about representativity: Suppose fw 3. Then for any vertex v, consider the faces incident with v, f 1, f 2,..., f k. Then the above implies there are no more intersections between the faces; i.e. all the vertices and faces as pictured in Figure 13 are distinct. This is called a wheel neighborhood. Fact. fw 3 every vertex has a wheel neighborhood. Similarly, G has representativity at least 4 if and only if the analagous statement holds for every face (all the vertices are distinct or else there exists a non-null-homotopic closed curve intersecting G three times). See Figure 14. Example. The equivalent condition for face width at least 5 is (loosely) that for every vertex there are two cycles of distinct faces surrounding it. In particular, this shows that the 5-grid embedded in P above has face width at least 5. 5

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