Graphs with no short cycle covers Edita M a cajov a Comenius - - PowerPoint PPT Presentation

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Graphs with no short cycle covers Edita M a cajov a Comenius - - PowerPoint PPT Presentation

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Graphs with no short cycle covers Edita M a cajov a Comenius University, Bratislava Ghent, August 2019 joint work with Martin Skoviera Edita M a cajov a (Bratislava) Lower bound od SCC August 2019 1 / 23 Cycle cover

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Graphs with no short cycle covers

Edita M´ aˇ cajov´ a

Comenius University, Bratislava Ghent, August 2019

joint work with Martin ˇ Skoviera

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 1 / 23

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Cycle cover

cycle – a graph with every vertex of even degree

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 2 / 23

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Cycle cover

cycle – a graph with every vertex of even degree cycle cover of a bridgeless graph G – a collection of cycles that cover every edge of G

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 2 / 23

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Cycle cover

cycle – a graph with every vertex of even degree cycle cover of a bridgeless graph G – a collection of cycles that cover every edge of G length of a cycle cover C – the sum of lengths of all the cycles in C

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 2 / 23

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Cycle cover

cycle – a graph with every vertex of even degree cycle cover of a bridgeless graph G – a collection of cycles that cover every edge of G length of a cycle cover C – the sum of lengths of all the cycles in C scc(G) ... the length of a shortest cycle cover

Short cycle cover problem (Itai, Rodeh, 1978)

Given a bridgeless graph, what is the length of its shortest cycle cover? quickly gained great prominence

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 2 / 23

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Cycle cover

cycle – a graph with every vertex of even degree cycle cover of a bridgeless graph G – a collection of cycles that cover every edge of G length of a cycle cover C – the sum of lengths of all the cycles in C scc(G) ... the length of a shortest cycle cover

Short cycle cover problem (Itai, Rodeh, 1978)

Given a bridgeless graph, what is the length of its shortest cycle cover? quickly gained great prominence

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 2 / 23

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Short cycle cover conjecture

Short cycle cover conjecture (Alon, Tarsi; Jaeger; 1985)

Every bridgeless graph G has a cycle cover of length at most 7

5 · |E(G)|.

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 3 / 23

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SLIDE 8

Short cycle cover conjecture

Short cycle cover conjecture (Alon, Tarsi; Jaeger; 1985)

Every bridgeless graph G has a cycle cover of length at most 7

5 · |E(G)|.

scc(Pg) = 7

5 · |E(Pg)|

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 3 / 23

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Short cycle cover conjecture

Short cycle cover conjecture (Alon, Tarsi; Jaeger; 1985)

Every bridgeless graph G has a cycle cover of length at most 7

5 · |E(G)|.

scc(Pg) = 7

5 · |E(Pg)|

Theorem (Bermond,Jackson,Jaeger 1983; Alon, Tarsi, 1985)

Every bridgeless graph G has a cycle cover of length at most 5

3 · |E(G)|.

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 3 / 23

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Related problems

Conjecture is optimization in nature, however it implies Cycle double cover conjecture (Jamshy, Tarsy, 1992)

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 4 / 23

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Related problems

Conjecture is optimization in nature, however it implies Cycle double cover conjecture (Jamshy, Tarsy, 1992) SCCC is implied by the Petersen colouring conjecture

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 4 / 23

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Related problems

Conjecture is optimization in nature, however it implies Cycle double cover conjecture (Jamshy, Tarsy, 1992) SCCC is implied by the Petersen colouring conjecture Chinese postman problem scc(G) ≥ cp(G)

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 4 / 23

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SCCC and cubic graphs

crucial are cubic graphs because the largest values of the covering ratio between scc(G) and |E(G)| are known for cubic graphs

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 5 / 23

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SCCC and cubic graphs

crucial are cubic graphs because the largest values of the covering ratio between scc(G) and |E(G)| are known for cubic graphs the covering ratio 7

5 is reached for infinitely many cubic graphs with

cyclic connectivity 2 and 3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 5 / 23

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SCCC and cubic graphs

crucial are cubic graphs because the largest values of the covering ratio between scc(G) and |E(G)| are known for cubic graphs the covering ratio 7

5 is reached for infinitely many cubic graphs with

cyclic connectivity 2 and 3 [Fan 2017] the covering ratio for bridgeless cubic graphs is at most 218/135 (≈ 1.6148)

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 5 / 23

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SCCC and cubic graphs

crucial are cubic graphs because the largest values of the covering ratio between scc(G) and |E(G)| are known for cubic graphs the covering ratio 7

5 is reached for infinitely many cubic graphs with

cyclic connectivity 2 and 3 [Fan 2017] the covering ratio for bridgeless cubic graphs is at most 218/135 (≈ 1.6148) [Lukot ’ka 2017] the covering ratio for bridgeless cubic graphs is at most 212/135 (≈ 1.5703)

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 5 / 23

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SCCC and cubic graphs

a natural lower bound for the covering ratio of cubic graphs is 4

3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 6 / 23

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SCCC and cubic graphs

a natural lower bound for the covering ratio of cubic graphs is 4

3

3-edge-colourable cubic graphs have the covering ratio 4

3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 6 / 23

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SCCC and cubic graphs

a natural lower bound for the covering ratio of cubic graphs is 4

3

3-edge-colourable cubic graphs have the covering ratio 4

3

all cyclically 4-edge-connected cubic graphs where the covering ratio is known have the value close to 4

3 [Brinkmann, Goedgebeur,

H¨ agglund, Markstr¨

  • m, 2013]

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 6 / 23

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SCCC and cubic graphs

a natural lower bound for the covering ratio of cubic graphs is 4

3

3-edge-colourable cubic graphs have the covering ratio 4

3

all cyclically 4-edge-connected cubic graphs where the covering ratio is known have the value close to 4

3 [Brinkmann, Goedgebeur,

H¨ agglund, Markstr¨

  • m, 2013]

[Brinkmann, Goedgebeur, H¨ agglund, Markstr¨

  • m, 2013] up to 36

vertices there are two non-trivial cubic graphs that have the covering ratio greater than 4

3 (the Petersen graph and G34 discovered by

H¨ agglund in 2016)

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 6 / 23

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SCCC and cubic graphs

a natural lower bound for the covering ratio of cubic graphs is 4

3

3-edge-colourable cubic graphs have the covering ratio 4

3

all cyclically 4-edge-connected cubic graphs where the covering ratio is known have the value close to 4

3 [Brinkmann, Goedgebeur,

H¨ agglund, Markstr¨

  • m, 2013]

[Brinkmann, Goedgebeur, H¨ agglund, Markstr¨

  • m, 2013] up to 36

vertices there are two non-trivial cubic graphs that have the covering ratio greater than 4

3 (the Petersen graph and G34 discovered by

H¨ agglund in 2016) both these graphs have scc(G) = 4

3 · |E(G)| + 1

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 6 / 23

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SCCC and cubic graphs

a natural lower bound for the covering ratio of cubic graphs is 4

3

3-edge-colourable cubic graphs have the covering ratio 4

3

all cyclically 4-edge-connected cubic graphs where the covering ratio is known have the value close to 4

3 [Brinkmann, Goedgebeur,

H¨ agglund, Markstr¨

  • m, 2013]

[Brinkmann, Goedgebeur, H¨ agglund, Markstr¨

  • m, 2013] up to 36

vertices there are two non-trivial cubic graphs that have the covering ratio greater than 4

3 (the Petersen graph and G34 discovered by

H¨ agglund in 2016) both these graphs have scc(G) = 4

3 · |E(G)| + 1

[Esperet, Mazzuoccolo, 2014] infinite family with scc(G) > 4

3 · |E(G)|

[Esperet, Mazzuoccolo, 2014] there exists G with scc(G) ≥ 4

3 · |E(G)| + 2

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 6 / 23

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SCCC and cubic graphs

Conjecture (Brinkmann, Goedgebeur, H¨ agglund, Markstr¨

  • m, 2013)

For every cyclically 4-edge-connected cubic graph G with m edges scc(G) ≤ 4 3 · m + o(m).

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 7 / 23

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SCCC and cubic graphs

Conjecture (Brinkmann, Goedgebeur, H¨ agglund, Markstr¨

  • m, 2013)

For every cyclically 4-edge-connected cubic graph G with m edges scc(G) ≤ 4 3 · m + o(m). evidence for this conjecture: [H¨ agglund, Markstr¨

  • m, 2013], [Steffen, 2015]

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 7 / 23

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SCCC and cubic graphs

Conjecture (Brinkmann, Goedgebeur, H¨ agglund, Markstr¨

  • m, 2013)

For every cyclically 4-edge-connected cubic graph G with m edges scc(G) ≤ 4 3 · m + o(m). evidence for this conjecture: [H¨ agglund, Markstr¨

  • m, 2013], [Steffen, 2015]

we disprove the conjecture:

Theorem (EM,ˇ Skoviera)

There exists a family of cyclically 4-edge-connected cubic graphs Gn, n ≥ 1 such that scc(Gn) ≥ (4 3 + 1 69)|E(Gn)|.

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 7 / 23

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SCC and pmi

G – bridgeless cubic graph pmi(G) = the minimum number of perfect matchings that cover all the edges of G

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 8 / 23

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SCC and pmi

G – bridgeless cubic graph pmi(G) = the minimum number of perfect matchings that cover all the edges of G [H¨ agglund, Markstr¨

  • m, 2013; Steffen 2015] if scc(G) > 4

3 then

pmi(G) = 5

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 8 / 23

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Sketch of the proof

weight of an edge e in C – the number of cycles containing e

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 9 / 23

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Sketch of the proof

weight of an edge e in C – the number of cycles containing e weight of a vertex v in C – sum of weights of edges adjacent to v

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 9 / 23

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Sketch of the proof

weight of an edge e in C – the number of cycles containing e weight of a vertex v in C – sum of weights of edges adjacent to v weight of a multipole – sum of weights of its dangling edges

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 9 / 23

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Sketch of the proof

weight of an edge e in C – the number of cycles containing e weight of a vertex v in C – sum of weights of edges adjacent to v weight of a multipole – sum of weights of its dangling edges for each cubic graph G, scc(G) ≥ 4

3 · |E(G)|

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 9 / 23

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Sketch of the proof

weight of an edge e in C – the number of cycles containing e weight of a vertex v in C – sum of weights of edges adjacent to v weight of a multipole – sum of weights of its dangling edges for each cubic graph G, scc(G) ≥ 4

3 · |E(G)|

a cubic graph G such that scc(G) = 4

3 · |E(G)| is called light

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 9 / 23

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Resistant (2, 2)-pole

no light cover

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 10 / 23

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Resistant (2, 2)-pole

no light cover resistant (2,2)-pole

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 10 / 23

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Properties of a resistant (2, 2)-pole

S(e) – set of cycles containing e

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 11 / 23

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Properties of a resistant (2, 2)-pole

S(e) – set of cycles containing e

Lemma

M = resistant (2, 2)-pole

f g h M e

In every light cover if w(M) = 4, then S(e) = S(f ) and S(g) = S(h) if M is of type C then S(e) ∩ S(f ) = ∅ and S(g) ∩ S(h) = ∅ if w(e) = w(f ) = 1 and e and f belong to the different cycles of the cycle cover, then w(g) = w(h) = 2

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 11 / 23

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(2, 2)-pole Z

X1 X2 X3 X4

X1, X2, X3, X4 – resistant (2, 2)-poles

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 12 / 23

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Z is not a light (2, 2)-pole

suppose to the contrary that Z is light

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 13 / 23

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Z is not a light (2, 2)-pole

suppose to the contrary that Z is light then

◮ each vertex has weight 4 ◮ each edge is in 1 or 2 cycles Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 13 / 23

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Z is not a light (2, 2)-pole

suppose to the contrary that Z is light then

◮ each vertex has weight 4 ◮ each edge is in 1 or 2 cycles

for i ∈ {1, 2, 3, 4}, the weight of each Xi is 4, 6, or 8

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 13 / 23

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Z is not a light (2, 2)-pole

suppose to the contrary that Z is light then

◮ each vertex has weight 4 ◮ each edge is in 1 or 2 cycles

for i ∈ {1, 2, 3, 4}, the weight of each Xi is 4, 6, or 8 for i ∈ {2, 3}, the weight of each Xi is 4 or 6

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 13 / 23

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Z is not a light (2, 2)-pole

suppose that weitht of Xi is 8 for i ∈ {2, 3}

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 14 / 23

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Z is not a light (2, 2)-pole

suppose that weitht of Xi is 8 for i ∈ {2, 3}

Xi

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 14 / 23

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Z is not a light (2, 2)-pole

suppose that weitht of Xi is 8 for i ∈ {2, 3}

Xi

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 14 / 23

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Z is not a light (2, 2)-pole

suppose that weitht of Xi is 8 for i ∈ {2, 3}

Xi

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 14 / 23

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SLIDE 46

Z is not a light (2, 2)-pole

suppose that weitht of Xi is 8 for i ∈ {2, 3}

Xi

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 14 / 23

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Z is not a light (2, 2)-pole

suppose that weitht of Xi is 8 for i ∈ {2, 3}

Xi

a contradiction

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 14 / 23

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Types of X2 and X3

each of X2 and X3 is of one of the types A, L, R, C

type A Xi Xi Xi type L type R type C Xi

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 15 / 23

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Classification by (α, β) = (type of X2, type of X3)

X1 X2 X3 X4

α, β ∈ {A, C, L, R} (16 possibilities)

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 16 / 23

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(α, β) ∈ {(L, R), (L, C), (C, R)}

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 17 / 23

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(α, β) ∈ {(L, R), (L, C), (C, R)}

X2 X3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 17 / 23

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(α, β) ∈ {(L, R), (L, C), (C, R)}

X2 X3

a contradiction

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 17 / 23

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(α, β) ∈ {(A, A), (A, L), (R, A), (R, L)}

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 18 / 23

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(α, β) ∈ {(A, A), (A, L), (R, A), (R, L)}

X2 X3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 18 / 23

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(α, β) ∈ {(A, A), (A, L), (R, A), (R, L)}

X2 X3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 18 / 23

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(α, β) ∈ {(A, A), (A, L), (R, A), (R, L)}

X2 X3

a contradiction

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 18 / 23

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(α, β) ∈ {(L, L), (R, R)}

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 19 / 23

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(α, β) ∈ {(L, L), (R, R)}

X2 X3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 19 / 23

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(α, β) ∈ {(L, L), (R, R)}

X2 X3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 19 / 23

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(α, β) ∈ {(L, L), (R, R)}

X2 X3

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 19 / 23

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(α, β) ∈ {(L, L), (R, R)}

X2 X3

a contradiction

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 19 / 23

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we exclude many other ordered pairs (α, β) for α, β ∈ {A, C, L, R}

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 20 / 23

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we exclude many other ordered pairs (α, β) for α, β ∈ {A, C, L, R} (α, β) ∈ {(C, C), (R, C), (C, L)} and w(K) = 8

K X2 X3 X4 X1

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 20 / 23

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we exclude many other ordered pairs (α, β) for α, β ∈ {A, C, L, R} (α, β) ∈ {(C, C), (R, C), (C, L)} and w(K) = 8

K X2 X3 X4 X1

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 20 / 23

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we exclude many other ordered pairs (α, β) for α, β ∈ {A, C, L, R} (α, β) ∈ {(C, C), (R, C), (C, L)} and w(K) = 8

K X2 X3 X4 X1

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 20 / 23

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SLIDE 66

we exclude many other ordered pairs (α, β) for α, β ∈ {A, C, L, R} (α, β) ∈ {(C, C), (R, C), (C, L)} and w(K) = 8

K X2 X3 X4 X1

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 20 / 23

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SLIDE 67

we exclude many other ordered pairs (α, β) for α, β ∈ {A, C, L, R} (α, β) ∈ {(C, C), (R, C), (C, L)} and w(K) = 8

K X2 X3 X4 X1

a contradiction

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 20 / 23

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SLIDE 68

we exclude many other ordered pairs (α, β) for α, β ∈ {A, C, L, R} (α, β) ∈ {(C, C), (R, C), (C, L)} and w(K) = 8

K X2 X3 X4 X1

a contradiction ... the (2, 2)-pole Z is not light

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 20 / 23

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SLIDE 69

.

the (2, 2)-pole Z is not light

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 21 / 23

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SLIDE 70

.

the (2, 2)-pole Z is not light let Gk be the cyclic junction of k copies of Z

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 21 / 23

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SLIDE 71

.

the (2, 2)-pole Z is not light let Gk be the cyclic junction of k copies of Z in every cycle cover of Gk at least k vertices have weight at least 6

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 21 / 23

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.

the (2, 2)-pole Z is not light let Gk be the cyclic junction of k copies of Z in every cycle cover of Gk at least k vertices have weight at least 6 all vertices have weight at least 4

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 21 / 23

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SLIDE 73

.

the (2, 2)-pole Z is not light let Gk be the cyclic junction of k copies of Z in every cycle cover of Gk at least k vertices have weight at least 6 all vertices have weight at least 4 scc(Gk) ≥ ( 4

3 + 1 69)|E(Gk)|

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 21 / 23

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SLIDE 74

Conclusion

Conjecture (Brinkmann, Goedgebeur, H¨ agglund, Markstr¨

  • m, 2013)

For every cyclically 4-edge-connected cubic graph G with m edges, scc(G) ≤ 4 3 · m + o(m).

Conjecture (EM, ˇ Skoviera)

For every cyclically 5-edge-connected cubic graph G with m edges, scc(G) ≤ 4 3 · m + o(m).

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 22 / 23

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SLIDE 75

Thank you for your attention!

Edita M´ aˇ cajov´ a (Bratislava) Lower bound od SCC August 2019 23 / 23