Directed cycle double covers and cut-obstacles

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Directed cycle double covers and cut-obstacles

Andrea Jim´enez

University of S˜ao Paulo

Joint work with Martin Loebl

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DCDC conjecture

Consider an undirected graphG

Cycle double cover(CDC): set of cycles ofGcovering edges exactly twice Directed cycle double cover(DCDC): CDC + orientations

DCDC Conjecture [Jaeger – 1985]

Every bridgeless graph has a DCDC. CDC Conjecture [Seymour, Szekeres∼1975?]

Every bridgeless graph has a CDC.

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DCDC conjecture

Cycle double cover(CDC): set of cycles ofGcovering edges exactly twice

Directed cycle double cover(DCDC): CDC + orientations

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DCDC conjecture

Cycle double cover(CDC): set of cycles ofGcovering edges exactly twice

Directed cycle double cover(DCDC): CDC + orientations

CDC Conjecture [Seymour, Szekeres∼1975?]

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DCDC conjecture

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DCDC conjecture

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DCDC conjecture

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Related work

minimum counterexample to the DCDC conjecture: cubic, cyclically 4-edge-connected, non 3-edge-colorable.

DCDC conjecture holds in:

b bridgeless planar graphs,

b graphs with a nowhere-zero 4-flow(Jaeger, Tutte,≤1979), b 2-connected projective-planar graphs(Ellingham & Zha, 2011), b lean fork graphs(J. & Loebl, 2013+).

Topological approach to the DCDC conjecture (Jaeger, 1979).

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Trigraphs

AtrigraphH is a cubic graph withear decomposition H0, L1. . . , Ln

s.t.H0 is a cycle,Liis ak-ear withk∈ {1,2,3}or a star.

ear := star or path

k-ear := path withk internal vertices

Example:

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Trigraphs

ear := star or path

k-ear := path withk internal vertices Example:

1−ear

starting cycle 3−ear

2−ear star

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Trigraphs

ear := star or path

1−ear starting cycle

3−ear 2−ear star

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Trigraphs

ear := star or path

1−ear starting cycle

3−ear 2−ear star

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Trigraphs

ear := star or path

3−ear 2−ear star

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Trigraphs

ear := star or path

3−ear 2−ear star

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Trigraph conjecture

Every robust trigraph admits a reduction process avoiding cut-obstacles.

Theorem [J. & Loebl, 2014+]

Our “trigraph conjecture” implies general DCDC conjecture.

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Trigraph conjecture

Reduction process and cut-obstacles:

Mixed graph(V, E, A, R) R:=forbidden pairs of arcs in a DCDC Proposition

In general, reductions of stars, 1-ears and 2-ears always exist!!

In robust trigraphs (because of definition of robust), reductions of 3-ears always exist!!

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Trigraph conjecture

1 2 3 4

5

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7

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Trigraph conjecture

1 2 3 4

5

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7

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Trigraph conjecture

1 2 3 4

5

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7

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Trigraph conjecture

1 2 3 4

5

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7

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Trigraph conjecture

1 2 3 4

5

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Trigraph conjecture

1 2 3 4

5

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Mixed graph(V, E, A, R) R:=forbidden pairs of arcs in a DCDC

Proposition

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Trigraph conjecture

1 2 3 4

5

Proposition

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Trigraph conjecture

1 2 3 4

5

Proposition

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Trigraph conjecture

1 2 3 4

5

Proposition

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Trigraph conjecture

1 2 4

5

1 2 3 4

Proposition

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Trigraph conjecture

1 2 3 4

Proposition

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Trigraph conjecture

1 2 3

Proposition

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Trigraph conjecture

1 2 3

Proposition

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Trigraph conjecture

1 2 3

Proposition

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Trigraph conjecture

1 2 3

Proposition

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Trigraph conjecture

1 2

Proposition

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Trigraph conjecture

1

Proposition

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Trigraph conjecture

Proposition

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Trigraph conjecture

Proposition

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Weak trigraph conjecture

In the closure of each robust trigraph, there exists a trigraph that admits a reduction process avoiding cut-obstacles.

The “weak trigraph conjecture” implies general DCDC conjecture.

Closure of a robust trigraph

Given trigraphH,H. A trigraphH⁰,H⁰is in the closure ofH,HifH⁰,H⁰is obtained fromH,Hby a sequence of the following two operations:

Modification ofHup to starting cycle and a fixed subsetS of ears. A local exchange of edges (H⁰is not necessarily isomorphic toH). Proposition

Our conjecture holds in the class ofplanar trigraphsand for trigraphs that admit specialembeddings into orientable surfaces.

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Modification ofHup to starting cycle and a fixed subsetS of ears.

A local exchange of edges (H⁰ is not necessarily isomorphic toH).

Proposition

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Modification ofHup to starting cycle and a fixed subsetS of ears.

A local exchange of edges (H⁰ is not necessarily isomorphic toH).

Proposition

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Sketch of Proof

“Trigraph conjecture implies general DCDC conjecture”.

ConsiderGbridgeless cubic graph & an ear decomposition ofG. Proposition

Complete reduction process ofGconstructs a DCDC ofG. Main Lemma (Construction)

There exists a trigraphH(G) such that:

Reduction process ofH(G) without cut-obstacles encodes complete reduction process ofG.

IfGis 3-connected, thenH(G) is a robust trigraph. Key ingredient in the construction ofH(G) are: basic gadgets

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Sketch of Proof

ConsiderGbridgeless cubic graph & an ear decomposition ofG.

Proposition

Complete reduction process ofGconstructs a DCDC ofG.

Main Lemma (Construction)

IfGis 3-connected, thenH(G) is a robust trigraph. Key ingredient in the construction ofH(G) are: basic gadgets

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Sketch of Proof

Proposition

IfGis 3-connected, thenH(G) is a robust trigraph.

Key ingredient in the construction ofH(G) are: basic gadgets

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Sketch of Proof

Proposition

v

BASIC GADGET

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Directed cycle double covers and cut-obstacles