LR Structures from Cycle Structures Tale, Slovakia 1 July, 2009

1. LR Structuresfrom Cycle StructuresTale, Slovakia1 July, 2009 Joint work with Primoz Potocnik

2. A cycle decomposition of a graph L is a partition C of its edges into cycles: Three squares Four triangles

3. We say that (L, C) is bipartite provided that the cycles in C can be separated into two colors, R and G, so that every vertex belongs to one cycle from G and one from R. Five red 4-cycles and four green 5-cycles (on the torus)

4. An LR structure is a special kind of cycle decomposition which has three properties: (1) It is bipartite. The group of color-preserving symmetries is transitive on vertices.

5. (3) At every vertex v, there are red and green swappers: These are symmetries which fix (pointwise) one cycle at v, while reversing the other.

6. There are two ways in which an LR structure can be too tame to be interesting: (1) It might contain a 4-cycle which alternates color. This can happen only on the torus. (2) It might be self-dual: i.e., some symmetry of L preserves C but interchanges R and G.

7. An LR structure with neither of these properties is asuitableLR structure. Suitable LR structures correspond 1-1 to worthy semisymmetric graphs of girth and valence 4. Are there any?

8. Here’s a way to construct one: Start with a cycle decomposition D whose group is transitive on its darts: Such a D is called a cycle structure.

9. Next, pull it apart: Then join each new pair with two new edges: That means: divide each vertex into two vertices, one in each cycle

10. Assign voltages (mod 2): Voltage 0 Voltage 1

11. CS(D ,0) and CS(D ,1) are the LR structures given by the 2-covering specified by those voltage assignments. Swappers lift in both cases. Both LR structures are suitable. So it is possible to construct suitable LR structures. Why is that important?

12. The Partial Line Graph of a cycle decomposition: (1) One new vertex for each edge (2) Join two when the corresponding edges share a vertex but are not on the same cycle.

13. The Partial Line Graph of a suitable LR structure is a semisymmetric graph. Every worthy tetravalent semisymmetric graph of girth 4 is the Partial Line Graph of a suitable LR structure.

14. The smallest example of a cycle structure is this one:

15. For this cycle structure D, the partial line graphs of CS(D ,0) and CS(D ,1) are the smallest worthy tetravalent semisymmetric graphs. Moreover, these two are believed to have the smallest group (order 80) of any semisymmetric graphs of any degree.

16. Almost every tetravalent dart-transitive graph has at least one cycle structure. Most have 2 or 3. Graph Cycle structuresK5 two 5-cyclesOctahedron three 4-cycles, four 3-cyclesK4, 4 four 4-cycles, two 8-cyclesC3xC3 six 3-cycles, three 6-cyclesW(5,2) five 4-cyclesC10(1,3) two 10-cyclesW(6,2) six 4-cycles, two 12-cycles, four 6-cycles (two ways)R6(1, 2) eight 3-cycles, six 4-cycles four 6-cycles

18. Laymen think that mathematicians are people who are never confused. Actually, we are very often confused. We’re just more comfortable with it.