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

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

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Definition

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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.

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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)

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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

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Decompositions of pseudographs

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Irregular coloring - irregular number

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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]:

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Correspondence ? { }

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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

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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 ).

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

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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.

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Application - even - even

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Exception

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Decompositions of complete bipartite digraphs and even complete bipartite multigraphs

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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.

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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

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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

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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

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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 )

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

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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

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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

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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

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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)

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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

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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

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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.

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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.

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Problem

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

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

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

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Thank you very very much!!

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