Spanning Cubic Graph Designs for Decomposing Complete Graphs

1. Spanning Cubic Graph Designs for Decomposing Complete Graphs ANEESH HARIHARAN (AMATH), UNIVERSITY OF WASHINGTON, SEATTLE MOSHE ROSENFELD (MATH), UNIVERSITY OF WASHINGTON, SEATTLE

2. Spanning Graph Designs- a natural extension of BIBD’s • Interested in finding decomposition of the complete graph Kn. • A graph G decomposes the complete graph Kn if edges of Kn can be covered by edge disjoint copies of G • Blocks = Sub-graphs (G) • (n,G,1) – Graph G decomposes Kn. • If G has n vertices, spanning Graph Design.

3. Spanning 1-regular and Spanning 2-regular Graph Designs • Spanning 1-regular decomposition • Hamiltonian Cycle Decomposition (connected 2-regular decomposition) • K2k+1 can always be decomposed into k Hamilton cycles. • Oberwolfach Problem + + = K4 G0 G1 G2 a a e d c b = + c e d b

4. Problem Definition • Spanning 3-regular graphs that decompose Kn – received only sporadic interest. (both theoretical and applied) • Are there spanning cubic graphs that decompose Kn? In other words, can we pack edge disjoint copies of cubic graphs of order n in Kn? • Problem makes sense only when n=6k+4.

5. Computational Results • Imrich (1971) - There are 21 distinct cubic graphs of order 10 and • Adams et al. (1997) - 15 of them decompose K10 while 6 do not. Figure: The six graphs that do not decompose K10. • Each of them was individually shown as to why they do not decompose K10.

6. Proof of why the Peterson Graph does not decompose K10 • The Peterson Graph and an isomorphic form in its ‘partite like’ structure without the standard labels. • Note: The above two ‘partite structures’ when superimposed are edge disjoint. Hence two copies of Peterson fit in K10. g b(f) f f a(g) g i b d e` = = h(a) a c h ∞ ∞ e(c) f(b) b c ∞ a g(e) h e d(h) i(i) d i c(d)

7. Proof of why Peterson does not decompose K10.(contd..) • The remaining 15 edges: (a-f), (a-e), (g-e), (f-b), (f-d), (a-c), (g-c), (b-c), (g-i), (h-b), (d-h), (d-i), (e-∞), (h-∞), (i-∞) BIPARTITE !!!! , (Check: λ1=3, λ10= -3 (both multiplicity =1), whereas Peterson is not bipartite. Therefore 3 copies of Peterson cannot pack into K16. QED. e ∞ a h i b g c d f

8. Computational Results (Contd.) • Adams, Rosenfeld et al. (2009) – Almost all cubic graphs of order 16 decompose K16. Only 3 of the 4207 graphs did not decompose K16. Spanning cubic graphs that do not decompose K6n+4 seem rare. • The 3 graphs that do not decompose K16 (computational proofs only)are K3,3+K4+PS3, 2K4+G8, K3,3+Peterson. K3,3 K4 G8 PS3

9. Cubic graphs that decompose K6K+4: By 1-Factorization • One factorizations of Complete Graphs, K2m, introduced by Mendelsohn, Rosa (1985): M0=(∞,0),(1,2m-2),…..(m-1,m) Mk=(i+k,j+k) | (i,j) belongs to M0 for all k=1,2…..2m-2 with ∞+k=∞ and all arithmetic in mod(2m-1). • Consider Eg. G10 m=5,k=1. • Mappings Ω(x) = x+3 (mod 9) for x = 1…8 and Ω(∞)=∞, maps G0 to G1 to G2.

10. 1-Factorization to Cubic Graphs • Graph G0 = M0 U M1 U M2 • Graph G1 = M3 U M4 U M5 • Graph G2 = M6 U M7 U M8 • G0, G1,G2 – Pairwise edge disjoint . • Hence G0 decomposes K10.

11. The Standard Form Cubic Labeling • This form of labeling for Cubic Graphs was first introduced by Adams et al.(2009). 21 11 01 ∞ 02 03 12 13 22 23 Standard Cubic Labels for G0 that decomposes K10; n=1

12. The partite structure of cubic graphs: Standard Cubic Labels for G0 that decomposes K16 . 41 31 21 11 01 ∞ 02 03 12 13 23 22 32 33 43 42 Partite structure of cubic graph G16 that decomposes K16

13. Interchanging Edges on Standard Labels and Issue of Isomorphism • Consider the case of standard labels on the planar Hamiltonian cubic graph of order 22, that will decompose K22 (as proved previously) 01 13 52 22 43 41 31 53 12 62 Graph G0 ∞ 03 02 63 21 51 33 32 42 23 61 11 01 13 52 22 43 41 31 53 12 62 ∞ 03 Graph G1 Not isomorphic to G0 AND decomposes K22. 02 63 21 51 33 32 42 23 61 11 01 13 52 22 43 41 31 53 12 62 Graph G2 ; same λ2 as G0 ! COULD IT BE ISOMORPHIC TO G0 ∞ 03 02 63 21 51 33 32 42 23 61 11

14. Interchanging Edges on Standard Labels and Issue of Isomorphism G2 is indeed isomorphic to G0 !!!! 01 13 52 42 23 61 41 43 22 62 ∞ 03 02 63 21 31 53 12 32 33 51 11 Not isomorphic to G0 Summary: Standard Cubic Labels are great but SADLY isomorphism is definitely an issue. Is there a structural answer?

15. The n-Graceful Label • Dealing with Isomorphism is an entire problem by itself. • Assume ∞ is connected to 01,02 and 03. • Within a partite, is it possible to come up with edges in such a way so that in every single configuration each vertex in a partite set can be paired with every other vertex, so as to obey the standard cubic labeling rules within a partite set? • Consider K28. Assuming ∞ is connected to 01,02 and 03,,we get the following 7 possible configurations of vertices being matched, within a partite. • 1-2 1-3 1-4 1-5 1-6 1-7 1-8 4-6 2-5 2-6 2-3 3-4 2-4 2-7 5-8 4-8 7-8 6-8 2-8 5-6 3-6 3-7 6-7 3-5 4-7 5-7 3-8 4-5 • The above is the only unique such ‘Graceful Block’ for 1…8 in mod 9.

16. n-Graceful Block • This leads to the question: • Call an n-block graceful if:1. It consists of n pairwise disjoint pairs of distinct integers{xi,yi} taken from 1,2,...,2n. 2. {xi – yi, yi – xi, i = 1,...n} mod 2n+1 = {1,2,...,2n}.(In other words, all possible differences mod 2n+1 are distinct.)Question: Can the 2n choose 2 pairs from {1,2,...,2n} be partitionedinto 2n-1 graceful n-blocks for n>=4? • Unique? Don’t know. • Some of the blocks for n=7. 1-14 2-4 3-1 7-5 9-7 12-10 2-13 3-14 4-8 8-4 10-6 11-9 3-12 4-13 5-11 3-12 5-14 2-11 4-11 5-12 6-13 10-2 12-5 14-7 5-10 6-11 7-12 11-1 13-3 1-6 6-9 7-10 14-2 9-6 8-11 5-8 7-8 8-9 9-10 13-14 2-1 3-4

17. Application- Round Robin Scheduling under constraints • Assume there is multiple round robin tournament and we have to schedule matches. • Assume n teams that are playing have rankings 1…n and every team has to play every other team such that there are n/2 matches everyday, and each team plays exactly n-1 matches over n-1 days. Is there something we can do better than random scheduling? • Incorporating distinct difference constraints on each day can lead to a better ‘uniform’ scheduling that is consistent with time

18. Conclusion and Future Work Remarks: • Spanning Cubic Graph Designs is a field that has a long way to go, both mathematically and from an application standpoint. • A very good foundation in terms of labeling techniques has been laid that stems from the well known 1-factorization. • A structural Analysis of Isomorphism seems promising. • The Standard Cubic Labeling is a powerful technique and could have many potential applications. • The mathematical properties of large Graceful Blocks is unclear at the present, but could have potential applications for many real time scheduling problems (http://www.msri.org/publications/ln/msri/2000/combdes/dinitz/1/) Future Work: • Try to resolve the reason for graphs that do not decompose K16. • To see if the all cubic graphs have the general partite structure without the standard labels. • Validity of the Graceful Blocks, computationally (using a greedy approach) and mathematically. • Look at potential applications in fields of Network Security for strengthening key-predistribution schemes to avoid node capture and jamming due to adversaries.

19. References • Bondy and Murthy – Graph Theory with Applications • One Factorization of the Complete Graph – Mendelsohn, Rosa – J of Graph Theory 9 (1985), 43-65 • Spanning Cubic Graph Designs – Adams, Rosenfeld et al. – Discrete Mathematics 309 (2009) 5781-5788 • Cube Factorizations of Complete Graphs – Adams et al. – J combin. Des 12 (5) (2004) 381338 • Decomposing K10 into cubic factors with exactly two isomorphic components – Petrenjuk (preprint) • Dynamic Survey of Graph Labeling – Electronic Journal of Combinatorics 5, (2005), 1-148 • Graph Theory – Douglas West