1. Decompositions of graphs into closed trails of even sizes Sylwia Cichacz AGH University of Science and Technology, Kraków, Poland

2. Decompositions of pseudographs Decompositions of complete bipartite digraphs and even complete bipartite multigraphs Part 2 Part 3 Part 1 Part 4 Definition Problem

3. Definition

4. graph G of size ||G|| Decomposition sequence (t 1,...,t p ) there is a closed trail of length in (for all ). Df. 1. G is, arbitrarily decomposable into closed trails iff G can be edge-disjointly decomposed into closed trails (T 1,...,T p ) of lengths (t 1,...,t p ) resp.

5. graph G of size ||G||=12 Example sequence (6,6) sequence (4,4,4) there are closed trails of lengths 4,6,8 in G G sequence (4,8)

6. Observation If G is arbitrarily decomposable into closed trails, then G is eulerian. there is a closed trail of length 3 in K 4 K 4 can not be edge-disjointly decomposed into closed trails of lengths (3,3). K4K4

7. Decompositions of pseudographs

8. Irregular coloring - irregular number

9. Results 2-regular graph of size T.1. [M. Aigner, E. Triesch, Zs. Tuza, 1992]: T.2. [P. Wittmann, 1997]: - even T.3. [S. C., J. Przybyło, M. Woźniak, 2005]:

10. Correspondence ? { }

11. Results n is odd, or, or n is even, - irregular number for proper coloring T.4. [P.N. Balister, 2001]. Let Then we can write same subgraph of as an edge disjoint union of circuits of lengths iff either: T.5. [P.N. Balister, B. Bollobás, R.H. Schelp,2002] Let G be a 2 -regular graph of order n. Then

12. Results - even L.1. If, then we can edge-disjointly pack closed trails of lengths into. L.2. If, then is edge-disjointly decomposable into closed trails of lengths. - even The graph is edge-disjointly decomposable into closed trails of lengths iff: T.5. [M. Horňák, M. Woźniak, 2003]: - even there is a closed trail of length in (for all ).

13. Proof L.2. If, then is edge-disjointly decomposable into closed trails of lengths. Proof:

14. Proof L.1. If, then we can edge-disjointly pack closed trails of lengths into. L.2. If, then is edge-disjointly decomposable into closed trails of lengths.

15. Application - even - even

16. Exception

17. Decompositions of complete bipartite digraphs and even complete bipartite multigraphs

18. Definitions - digraph obtained from graph G by replacing each edge by the pair of arcs xy and yx. - multigraph where each edge xy occurs with multiplicity r.

19. Reminder T.4. [P.N. Balister, 2001]. Let Then we can write same subgraph of as an edge disjoint union of circuits of lengths iff either: n is odd, or, or n is even, then can be decomposed as edge-disjoint except in the case when n=6 and all T.7. [P.N. Balister, 2003] If union of directed closed trails of lengths of closed trails of lengths iff either Then can be written as edge-disjoint union T.8. [P.N. Balister, 2003] Assume a) r is even, or b) r and n are both odd and

20. into closed trails of lengths iff: T.9. The digraph is edge-disjointly decomposable Results there is a closed trail of length in (for all ).. The graph is edge-disjointly decomposable into closed trails of lengths iff: T.6. [M. Horňák, M. Woźniak, 2003] - even - even T.10. Let into closed trails of lengths iff: The multigraph is edge-disjointly decomposable r - odd a,b - even - even if if a=2 or b=2

21. we fix the number of the vertex set B and will argue on induction on a Proof Proof: into closed trails of lengths iff: T.9. The digraph is edge-disjointly decomposable - even

22. Proof into closed trails of lengths iff: T.9. The digraph is edge-disjointly decomposable - even (t 1,…t k )(t k+1,…t p )

23. Proof into closed trails of lengths iff: T.9. The digraph is edge-disjointly decomposable - even and - even v w w

24. into closed trails of lengths iff: T.9. The digraph is edge-disjointly decomposable - even The multigraph is edge-disjointly decomposable into closed trails of lengths iff: Observation 11. Let r be even. Results - even

25. Proof The multigraph is edge-disjointly decomposable into closed trails of lengths iff: Observation 11. Let r be even. - even Proof: we consider as an edge-disjoint union of and T.9. (digraphs) induction on r

26. T.10. Let a,b - even r - odd - even into closed trails of lengths iff: The multigraph is edge-disjointly decomposable if a=2 or b=2 if Proof

27. The multigraph is edge-disjointly decomposable T.10. Let a,b - even r - odd - even into closed trails of lengths iff: if a=2 or b=2 if Proof we consider as an edge-disjoint union of and Case 1. a=2 or b=2 Case 2. T.6. [ M. Horňák, M. Woźniak ] Ob.11. (for even multiplicity)

28. The multigraph is edge-disjointly decomposable T.10. Let a,b - even r - odd - even into closed trails of lengths iff: if a=2 or b=2 if Proof Case 1. a=2 or b=2 Let be the smallest integer such that for i=r+1,…,k

29. The multigraph is edge-disjointly decomposable T.10. Let a,b - even r - odd - even into closed trails of lengths iff: if a=2 or b=2 if Proof Case 1. a=2 or b=2 for i=r+1,…,k

30. The multigraph is edge-disjointly decomposable T.10. Let a,b - even r - odd - even into closed trails of lengths iff: if a=2 or b=2 if Proof Let be the smallest integer such that - even Case 2.

31. The multigraph is edge-disjointly decomposable T.10. Let a,b - even r - odd - even into closed trails of lengths iff: if a=2 or b=2 if Proof - even Case 2.

32. Problem

33. sequence (t 1,...,t p ), graph L n, n>2.

34. Problem sequence (t 1,...,t p ), graph L n, n>2. Necessity: IT IS NOT ENOUGH?

35. Problem sequence (t 1,...,t p ), graph L n, n>2. Necessity: Example (3, 3, 6, 6 )

36. Thank you very very much!!