Structure and hamiltonicity of 3-connected graphs with excluded minors

Transcription

1 E0 Structure and hamiltonicity of 3-connected graphs with excluded minors Mark Ellingham* Emily Marshall* Vanderbilt University, USA Kenta Ozeki National Institute of Informatics, Japan Shoichi Tsuchiya Tokyo University of Science, Japan * Supported by the Simons Foundation and the U. S. National Security Agency

2 E1 Hamiltonicity and planarity Whitney, 1931: All 4-connected planar triangulations are hamiltonian. Tutte, 1956: All 4-connected planar graphs are hamiltonian. We cannot reduce the connectivity: Herschel graph: 3-connected planar bipartite, nonhamiltonian.

3 E2 Hamiltonicity and planarity Whitney, 1931: All 4-connected planar triangulations are hamiltonian. Tutte, 1956: All 4-connected planar graphs are hamiltonian. We cannot reduce the connectivity even for triangulations: Reynolds triangulation, 1931 (alias Goldner- Harary graph): 3-connected planar triangulation, nonhamiltonian.

4 E3 3-connected planar graphs But some weakenings of hamiltonicity are true for 3-connected planar graphs: Barnette, 1966: they have a 3-tree (spanning tree of maximum degree 3; weakening of hamilton path = 2-tree). Gao and Richter, 1994: they have a 2-walk (spanning closed walk using each vertex at most 2 times; weakening of hamilton cycle = 1-walk). Chen and Yu, 2002: they have a cycle of length at least cn log 3 2. So what conditions can we add to make them hamiltonian? Results on 3-connected planar graphs may also be regarded as essentially results on 3-connected K 3,3 -minor-free graphs.

5 E4 Minors of graphs We say H is a minor of G if we can identify each vertex v of H with a connected subgraph C v in G; C u and C v are vertex-disjoint when u v; if uv is an edge of H, then there is some edge between C u and C v in G. We say G is H-minor-free if it does not have H as a minor.

6 E5 Excluding K 3,t Chen, Egawa, Kawarabayashi, Mohar and Ota, 2011: For 3 a t, a-connected K a,t -minor-free graphs have toughness at 2 least (t 1)(a 1)!. Corollary: Using result of Win, 1989, get that 3-connected K 3,t -minor-free graphs have a (t+1)-tree. Improved by Ota and Ozeki, 2012: A 3- connected K 3,t -minor-free graph has a (t 1)-tree a t-tree This is best possible. if t is even, and if t is odd. Chen, Yu and Zang, 2012: A 3-connected K 3,t -minor-free graph has a cycle of length at least α(t)n β (β does not depend on t).

7 E6 Excluding K 2,t Easy: 2-connected K 2,3 -minor-free implies K 4 or outerplanar, therefore hamiltonian. Chen, Sheppardson, Yu and Zang, 2006: 2-connected K 2,t -minor-free graphs have a cycle of length at least n/t t 1. Any result for 3-connected K 3,t -minor-free applies to 3-connected K 2,t -minor-free. Note that K 2,t -minor-free graphs are very sparse. Chudnovsky, Reed and Seymour, 2011: K 2,t - minor-free graphs have number of edges m (t+1)(n 1)/2.

8 E7 Excluding K 2,5 is not enough We have examples of 3-connected K 2,5 -minorfree graphs that are nonhamiltonian. But perhaps there are only finitely many.

9 E8 Planarity and excluding K 2,6 are not enough We have an infinite family of 3-connected K 2,6 -minor-free planar graphs that are nonhamiltonian. Herschel extends to

10 E9 Planar hamiltonicity result Theorem (E, Marshall, Ozeki and Tsuchiya): Every 3-connected K 2,5 -minor-free planar graph is hamiltonian.

11 E10 Proof: general setup Assume nonhamiltonian. Take longest cycle C and one component L of G V(C). L must be joined to C at v 1,v 2,...,v k, k 3. Each interval I j along C between v j, v j+1 must be nonempty, else longer cycle. By 3-connectivity, must be edges leaving the intervals.

12 E11 Proof: some typical situations Overall idea: case analysis, find minor or longer cycle. Minor from edges jumping between intervals. Minor from crossing edges inside intervals (rooted K 2,2 -minors form part of overall K 2,5 -minor).

13 E12 Excluding rooted K 2,2 -minors Lemma: Suppose x,y V(H) and H +xy is 2-connected. Then these are equivalent: (i) H has no K 2,2 -minor rooted at x and y. (ii) H is xy-outerplanar: it has a spanning xy-path and all other edges can be drawn in the plane on one side of that path.

14 E13 Dropping planarity 3-connected K 2,5 -minor-free nonplanar graphs are not necessarily hamiltonian. What about 3-connected K 2,4 -minor-free graphs? Not only are all 3-connected K 2,4 -minor-free graphs hamiltonian, but we can also give a complete structure theorem for them. Original proof examined structure relative to a longest nonhamiltonian cycle. We recently found a shorter proof using results of Ding and Liu. In fact we give a complete structure theorem for all K 2,4 -minor-free graphs, not just 3-connected ones.

15 E14 K 2,4 -minor-free graphs Dieng and Gavoille, conference talk, 2008: Every 2-connected K 2,4 -minor-free graph has a set of two vertices whose deletion leaves an outerplanar graph. Interested in K 2,4 -minor-free graphs for object location/routing problems. Streib and Young, preprint, 2010: If G is connected and K 2,4 -minor-free then the poset of (labelled) minors of G has dimension bounded by a polynomial in E(G). They conjecture this is true for K 2,t -minorfree graphs for each fixed t. Uses Dieng & Gavoille result.

16 E15 Connectivity of K 2,4 -minor-free Since K 2,4 is 2-connected, a graph is K 2,4 -minor-free if and only if every block is K 2,4 -minor-free. So we only need to consider 2-connected graphs. Every 4-connected graph (except K 5 ) has a K 2,4 -minor, by a simple application of Menger s Theorem. So we only need to consider connectivity 2 or 3. Strategy: characterize 3-connected first, then use that to characterize graphs with a 2-cut.

17 E16 A 3-connected family G n,r,s for r,s = 0,1,...,n 3 has path v 1 v 2...v n (spine), v 1 v n i for i = 1,2,...,r, v n v 1+i for i = 1,2,...,s. Add v 1 v n for G + n,r,s. Proposition: Take family G with wheel K 1 +C n 1, G n,r,s and G + n,r,s for r,s = 2,3,...,n 3 and r+s = n 2 or n 1 for each n 5. Then (i) All planar, 3-connected, K 2,4 -minor-free. (ii) 2n 8 nonisomorphic graphs of order n.

18 E17 Result for 3-connected Theorem (E, Marshall, Ozeki and Tsuchiya, 2012+): A 3-connected graph is K 2,4 -minorfree if and only if it is isomorphic to a graph in G or to one of ten small examples (two planar, rest nonplanar). K 4 = G + 4,1,1 K 5 K 3,3 K + 3,3 K +,+ 3,3 A A + D C C +

19 E18 Proof approaches for 3-connected Original approach: analyse structure relative to largest nonhamiltonian cycle. Like proof for 3-conn. K 2,5 -minor-free planar graphs. New approach: apply results on H-minorfree for small 3-connected H. G. Ding & C. Liu, 2013: Characterized 3- connected H-minor-free graphs for all 3- connected graphs H with at most 11 edges. We use H = two supergraphs of K 2,4 : K 3,3 Octahedron\e

20 E19 Proof details for 3-connected Ding & Liu: Suppose G is 3-connected and H-minor-free, with at least 11 vertices. (i) If H = K 3,3, then G is planar. (ii) If H = Octahedron\e, then G comes from gluing wheels and prisms K 2 K 3 along a single triangle T, and possibly deleting edges of T. Suppose G is 3-connected and K 2,4 -minor-free with at least 11 vertices. Steps: 1. Since both graphs H above have K 2,4 as a subgraph, G satisfies (i) and (ii). 2. From (ii) we glue together wheels and prisms. 3. Gluing more than two wheels or prisms gives a K 3,3 -minor, which is nonplanar, violating (i). 4. Situations with prisms give K 2,4 -minors. 5. So G is a wheel, or two wheels glued along a triangle, and we can show that G G.

21 E20 Analysis for 2-connected Take 2-cut {x,y} in K 2,4 -minor-free G, nontrivial xy-bridges G 1,G 2,...,G k. Then k < 4 and at most one G i has a K 2,2 -minor rooted at x,y. If k = 2 or 3 and no G i has a K 2,2 -minor rooted at x,y: G is union of k xy-outerplanar graphs, and possibly edge xy. For k = 2 this means G is outerplanar. Otherwise k = 2, G 1 has a K 2,2 -minor rooted at x,y and G 2 does not. Take maximal G 2, replace G 2 by edge. Repeat, eventually reduce G to unique 3-connected core.

22 E21 Result for 2-connected Theorem (E, Marshall, Ozeki and Tsuchiya): A 2-connected graph is K 2,4 -minor-free if and only if it is (i) outerplanar, or (ii) the union of three nontrivial xy-outerplanar graphs and possibly the edge xy, or (iii) a 3-connected K 2,4 -minor-free graph with edges in a subdividable set {x 1 y 1,x 2 y 2,..., x k y k } replaced by x i y i -outerplanar graphs. Subdividable set: subdivide all edges once, remains K 2,4 -minor-free. For most members of G, subdividable = subset of spine edges.

23 E22 Possible further hamiltonicity result Computational investigations: Looked for K 2,6 -minor-free instances of nonhamiltonian 3-connected planar graphs. ME & G. Royle: 11 to 14 vertices, G. Royle: 15 and 16 vertices. Results suggest structure is very restricted. Conjecture: Every 3-connected K 2,6 -minor-free planar graph G is hamiltonian unless G F. Here F is a family with 206 graphs of order at most 15, and with exactly 40 graphs of order n for all n 16.

24 E23 Future directions Is the number of nonhamiltonian 3-connected K 2,5 -minor-free general graphs finite or infinite? David Wood: Let N be the class of planar graphs G for which every minor of G is a subgraph of a hamiltonian planar graph. N is a minor-closed class. What are the minimal forbidden minors besides K 5 and K 3,3? (The essential nonhamiltonian planar graphs.) Use characterization to check previous results on K 2,4 -minor-free graphs. Can we characterize K 2,5 -minor-free planar graphs? Or even general graphs? Apply Ding & Liu s results to characterize H-minor-free graphs for other small 2- connected H.