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Small Cycle Systems

Small Cycle Systems. Chris Rodger Auburn University. Hamilton Cycle. 4-cycle. Graph Decompositions - Paths. Partition the edges of your favorite graph so that each element of the partition induces something interesting. Use colors on the edges to denote the partition.

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Small Cycle Systems

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  1. Small Cycle Systems Chris Rodger Auburn University

  2. Hamilton Cycle 4-cycle

  3. Graph Decompositions - Paths • Partition the edges of your favorite graph so that each element of the partition induces something interesting. Use colors on the edges to denote the partition. Maybe you like paths, but with variety! K7 =

  4. Graph Decompositions - Cycles • Partition the edges of your favorite graph so that each element of the partition induces something interesting. Use colors on the edges to denote the partition. Or perhaps you like small cycles. This is a C3-decomposition of K7. Steiner Triple System K7 =

  5. Existence Problems • When do 4-cycle systems exist? • Clearly the number of edges must be divisible by 4 • And the degree of each vertex must be even. • So n must be congruent to 1 modulo 8. 1 Each edge has an associated “difference”. 3 4 2

  6. One Construction • 4-cycle decompositions are easy to make now that we have a decomposition of K9. Let’s try K17 Put 4-cycle decompositions of K9 on each of these two sets of 9 vertices. Pair the vertices and place a 4-cycle between pairs in different columns.

  7. It Solves the Existence Problem Add as many columns as you like!

  8. Another 4-cycle system of K17 Sometimes we need something more complicated Rotate left to right Pure Difference 4 ∞ Mixed Difference 4 What’s missing?? Pure Difference 4 Mixed differences 2 and -2

  9. Other Cycle Lengths • Of course that was just for 4-cycles. • The existence of m-cycle systems of order n has been solved after a long history. • Clearly • the number of edges must be divisible by m • the degree of each vertex must be even, and • n must be at least m, orn= 1. Alspach, Gavlas, Šajna (Hoffman, Lindner, Rodger)

  10. Partial Cycle Systems • A set of edge disjoint 4-cycles in Kn is said to be a partial 4-cycle system of order n. This is a partial 4-cycle system of order 11 that is EQUITABLE.

  11. Equitable Partial Cycle Systems • Equitable: for each pair of vertices u and v, the number of cycles containing u differs by at most one from the number of cycles containing v. These are VERY useful! • 3-cycles: Andersen, Hilton and Mendelsohn • 4-cycles and 5-cycles: Raines and Staniszló • Any mixture of cycle lengths! Bryant, Horsley and Maenhaut

  12. Embeddings • A 4-cycle system P of λKv is said to be embedded in a 4-cycle system Q of λKv+u if P is a sub-multiset of Q. P with λ = 1 Q

  13. We Have An Embedding Already This is an embedding of a 4-cycle system of K9 (left vertices) into a 4-cycle system of K17.

  14. Embeddings - History • The Lindner problem of embedding a partial 3-cycle system of order n into an STS(v) has been solved! (Bryant and Horsley) • A necessary condition requires that v ≥ 2n+1 • For 4-cycles the situation is messier, but recent progress has been dramatic: • Necessarily v ≥ n+n1/2-1 • Lindner had the best result of 2n+15 until recently: • n + 121/2n3/4 + o(n3/4) (Lindner and Hilton) • n + n1/2 + o(n1/2) (Füredi and Lehel)

  15. Embeddings - History • Partial 3-cycle system of order n into an STS(v) • v ≥ 2n+1 (Bryant, Horsley) • Partial 4-cycles systems: • n + n1/2 + o(n1/2) (Füredi, Lehel) • Partial 2k-cycle systems • Around kn (Hoffman, Lindner ,Rodger) • Partial 2k+1-cycle systems • Around (4k+2)n (Lindner, Rodger, Stinson)

  16. Enclosings • A 4-cycle system P of λKv is said to be enclosed in a 4-cycle system Q of (λ+μ)Kv+u if P is a sub-multiset of Q. P with λ = 1 Q with λ+μ = 2

  17. Theorem A 4-cycle system of λK v can be enclosed in a 4-cycle system of (λ+μ)Kv+u if and only if • (v+u-1)(λ+μ) is even, • The number of new edges is divisible by 4, • If u = 1 then μ(v-1)/2 ≥ λ + μ, and • If u = 2 then μv(v-1)/2 ≥ λ + μ. (Newman and Rodger)

  18. The biggest case: u ≥ 3 Use existing results on maximum partial 4-cycle systems. Add at least 3 vertices It is easy to use the edges that join 2 new vertices! So let’s move on to the more interesting cases!

  19. The neatest case: u = 2 • Settling this case involves: • Equitable partial 4-cycle systems, • Directed Euler Tours, and • Expanding nearly-regular graphs into copies of K2,2. • But first we check the necessary condition.

  20. Recall: If u = 2 then μv(v-1)/2 ≥ λ + μ Add u = 2 vertices. Eventually μ more edges are used between each pair of vertices in here. So μv(v-1)/2 ≥ (λ + μ) There are λ + μ edges joining the 2 added vertices. In here λedges between each pair of vertices are already used.

  21. Sufficiency with u = 2: μv(v-1)/2 ≥ λ + μ So exactly μv(v-1)/2 – (λ + μ) edges “must” be in 4-cycles joining vertices in P. We just saw: there are λ + μ edges joining the 2 added vertices, each of which must be in 4-cycles like this. P In P, μedges between each pair of vertices need to be used in 4-cycles.

  22. Sufficiency with u = 2: μv(v-1)/2 ≥ λ + μ So start with an equitable partial 4-cycle system C1 of μKv with exactly μv(v-1)/2 – (λ + μ) edges! It turns out that this number is divisible by 4. P And it is not negative! In P, μedges between each pair of vertices need to be used in 4-cycles.

  23. Sufficiency with u = 2: μv(v-1)/2 ≥ λ + μ Now look at the complement in μKv of C1. It has exactly λ + μ edges! All vertices have even degree, so form a directed Euler tour. Start v End P Let C2 be the Set of these 4-cycles. In P, μedges between each pair of vertices ARE NOW used in 4-cycles. In P, μedges between each pair of vertices need to be used in 4-cycles.

  24. Sufficiency with u = 2: μv(v-1)/2 ≥ λ + μ The remaining edges induce a bipartite graph B between P and {S, E}. Since C1 is equitable, each vertex v in P has degree 2s or 2s+2 in B (for some s). Also note that s or s+1edges in B respectively join v to each of SandE. v Form a graph on V(P) in which each vertex v has degree dB(v)/2. S E P For each edge add a 4-cycle.

  25. The last necessary condition: If u = 1 then μ(v-1)/2 ≥ λ + μ Eventually μ more edges between pairs of vertices are used in here. So μv(v-1)/2 ≥ (λ + μ)v Add λ + μedges going to each of the v vertices. In here λedges between each pair of vertices are already used.

  26. A typical case:v = 4x+1; μ even (so 4 | λ+μ) ∞ is the added vertex 2i-1 2i 2i-1 2i 0 V-1 ∞ 2i-1 Select (λ+μ)/4 of the (μ/2)(v-1)/4 cells and form cycles like these. 2i 2i-1 2i Recall the necessary condition: μ(v-1)/2 ≥ λ + μ

  27. v = 4x+1; μ even 0 V-1 ∞ 2i-1 2i Select the remaining cells and form cycles like these. 2i 2i-1

  28. Another typical case:v ≡ 2 (mod 4); so 4 | λ,μ We have pure and mixed differences 1,2, …, (v-2)/4, and mixed difference 0. ∞ 0 λ/2 differences for these s and -s s s v/2 -1

  29. Another typical case:v ≡ 2 (mod 4); so 4 | λ,μ • We have pure and mixed differences 1,2, …, (v-2)/4, and mixed difference 0. ∞ 0 λ/2 differences for these s μ/4 differences for these This uses 4 times: mixed difference 0; ∞ to both sides s v/2 -1

  30. Another typical case:v ≡ 2 (mod 4); so 4 | λ,μ • We have pure and mixed differences 1,2, …, (v-2)/4, and mixed difference 0. ∞ 0 λ/2 differences for these μ/4 differences for these Use each of the remaining differences in these 4-cycles v/2 -1

  31. v ≡ 0 (mod 4) Really Tricky!Try μ≡ 3 (mod 4) • λ is even (v is λ-admissible) • So λ + μ is odd • 4 divides (λ + μ) v(v+1)/2 (v+1 is (λ+μ)-admissible) • So v ≡ 0 (mod 8) • μ(v-1)/2 = (4x+3)(4y+3)/2 = 4z + 1/2 • λ + μ ≤ μ(v-1)/2 - 3/2 So 1.5 parallel classes must avoid ∞!

  32. v ≡ 0 (mod 8); μ≡ 3 (mod 4) This vertex structure allows us to find the 1.5 parallel classes avoiding ∞ in a natural way. 1 copy of these (using: one v/8 difference; one mixed 0 difference; ∞ to both sides) The orange edges provide the required 1.5 parallel classes That avoid ∞ ∞ 0 v/8 v/4 v/2 -1

  33. Do you remember this? Sometimes we need something more complicated Rotate left to right

  34. Other Enclosings For 3-cycle systems: • The problem remains open. • There are earlier results by Colbourn and Hamm, and also with Rosa (this conference in 1985!). • There are several recent results by Hurd, Munson and Sarvate that consider small enclosings. • One of the necessary conditions is quadratic. • Enclosings do not exist in the interval: • Recent result approach this gap from both sides (Newman and Rodger Nothing appears to be known for longer cycles. (v+1)(1 –(1-(4mv)/(v-1)2(λ+m)2)1/2, (v+1)(1 +(1-(4mv)/(v-1)2(λ+m)2)1/2

  35. Spouse Avoiding Dinners Try to find a way for 4 couples to sit at 2 tables, each seating 4 people so that each sits next to each other person exactly once. Not the spouses! 1 2 3 4 Friday Saturday Men Women Sunday M2 M3 M1 W4 W4 W4 W2 W1 W3 M1 M2 M3 W2 W1 W3 W2 W3 W1 M4 M4 M4 M2 M1 M3

  36. Spouse Avoiding Dinners Try to find a way for 4 couples to sit at 2 tables, each seating 4 people so that each sits next to each other person exactly once. Not the spouses! 1 2 3 4 Friday Saturday Men Women Sunday Can you do this so that each table has 2 men and 2 women?

  37. Must one avoid one’s spouse?? No! You now have an excuse for another dinner! Cycle systems of graphs other than Kn are also interesting. Join vertices in the same group with λ1edges and vertices in different groups with λ2 edges λ1 = 2 and λ2 = 1 Pure andMixedEdges

  38. Cycle Systems with 2 Associate Classes Suppose a is even. There exists a C4-factorization of K(a,p;λ1,λ2) if and only if 4 divides ap λ1 is even, and Ifa ≡ 2 (mod 4) then λ2a(p-1) ≥ λ1. (Billington, Rodger) 1 2 λ1 a λ2 1 2 p K(a,p;λ1,λ2)

  39. Cycle Systems with 2 Associate Classes Suppose a≡ 1 (mod 4). There exists a C4-factorization of K(a,p;λ1,λ2) if and only if 4 divides p λ2> 0 and is even, and λ2a(p-1) ≥ λ1, (Rodger, Tiemeyer) 1 2 λ1 a λ2 1 2 p except possibly if a = 9 and λ1 is odd. K(a,p;λ1,λ2)

  40. What about a≡ 3 (mod 4)? Looks difficult from my point of view!

  41. Why must λ2a(p-1) ≥ λ1? Every part must contain at least 2 vertices incident with mixed edges. Suppose a≡ 2 (mod 4). Consider one C4-factor. So each C4-factor must contain at least p mixed edges! 1 2 a =6 K(a,p;λ1,λ2) 1 p

  42. 4-cycle systems of K(a,p;λ1,λ2) • There exists a 4-cycle system of K(a,p;λ1,λ2) if and only if • Each vertex has even degree, • The number of edges is divisible by 4, • If a = 2 then • λ2 > 0, and • λ1 ≤ 2(p-1) λ2 • If a = 3 then • λ2 > 0, and • λ1≤ 3(p-1) λ2/2 if λ2is even, and • λ1 ≤ 3(p-1) λ2/2 if λ2is odd. • (Hung Lin Fu, Rodger) • For 3-cycles: Fu, Rodger, Sarvate • For block designs: Bose and Shimamoto – 1952! - (p-1)/9

  43. Why is λ1≤ 3(p-1) λ2/2 when a = 3? • Every 4-cycle must use at least 2 mixed edges. • So3pλ1≤ 9p(p-1) λ2/2 K(a,p;λ1,λ2) Each of these uses an even number of mixed edges. 1 p

  44. Isλ1≤ 3(p-1) λ2/2 –(p-1)/9 when λ2 is odd? • There are an odd number of edges between each pair of parts! • So some 4-cycles must use at least 3 mixed edges • So3pλ1 ≤ 9p(p-1) λ2/2 – (p-1)/9 K(a,p;λ1,λ2) Each of these uses an even number of mixed edges. 1 p

  45. Plenty More! • 4-cycle systems that cover 2-paths (Heinrich and Nonay, Cox and Rodger, Kobayashi and Nakamura) • Resolvable versions (Kobayashi and Nakamura) • 4-cycle systems of line graphs of Kn and of line graphs of complete multipartite graphs (Rodger and Sehgal) • 4-cycle systems of Kn minus any graph with • maximum degree 3 • One vertex of any degree, all others of degree at most 2 (Fu, Fu and Rodger, Sehgal, Ash)

  46. Thanks for listening!

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