on matching coverings and cycle coverings · 2013-09-29 · circuit double cover conjecture is one...

On matching coverings and cycle coverings

Xinmin Hou (co-work with Hong-Jian Lai and Cun-Quan Zhang)

Email: [email protected]

School of of Mathematical Science

University of Science and Technology of ChinaHefei, Anhui 230026, China

X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 1 / 29

mailto:[email protected]

Contents

1 Definitions and Conjectures

2 A family of Berge coverable graphs

3 Matching coverings and cycle coverings


Definitions

Let 𝐺 be a graph.

A matching 𝑀 is a 1-regular subgraph of 𝐺. A perfect matching of 𝐺 is a

spanning 1-regular subgraph of 𝐺 (also called a 1-factor of 𝐺), and an 𝑟-factor of

𝐺 is a spanning 𝑟-regular subgraph of 𝐺.

A circuit is a connected 2-regular graph, and an even subgraph (also called a

cycle) is a subgraph such that each vertex has an even degree.

The suppressed graph, denote by 𝐺, is the graph obtained from 𝐺 by

suppressing all degree two vertices.


Matching coverings

A perfect matching covering ℳ of 𝐺 is a set of perfect matchings of 𝐺 if

every edge of 𝐺 is contained in at least one member of ℳ. Let 𝒯𝜇 be the set of

cubic graphs admitting perfect matching coverings ℳ with |ℳ| = 𝜇.

A perfect matching covering ℳ of 𝐺 is a (1, 2)-covering if every edge of 𝐺

is contained in precisely one or two members of ℳ. Let 𝒯 ⋆𝜇 be the set of cubic

graphs admitting perfect matching (1, 2)-coverings ℳ with |ℳ| = 𝜇.


Matching coverings

Let 𝒢 be the family of all bridgeless cubic graphs.

Conjecture 1.1 (Berge-Fulkerson Conjecture)

Every bridgeless cubic graph 𝐺 has a collection of six perfect matchings that

together cover every edge of 𝐺 exactly twice (or, equivalently, 𝒯 ⋆6 = 𝒢).

We call such a perfect matching covering in Conjecture 1.1 a Fulkerson

covering.

Conjecture 1.2 (Berge’s Conjecture)

Every bridgeless cubic graph 𝐺 has a collection of at most five perfect matchings

with the property that each edge of 𝐺 is contained in at least one member of

them (or, equivalently, 𝒯5 = 𝒢).

The perfect matching covering in Conjecture 1.2 is called a Berge covering

of 𝐺.


Matching coverings

Let 𝒢 be the family of all bridgeless cubic graphs.

Conjecture 1.1 (Berge-Fulkerson Conjecture)

Every bridgeless cubic graph 𝐺 has a collection of six perfect matchings that

together cover every edge of 𝐺 exactly twice (or, equivalently, 𝒯 ⋆6 = 𝒢).

We call such a perfect matching covering in Conjecture 1.1 a Fulkerson

covering.

Conjecture 1.2 (Berge’s Conjecture)

Every bridgeless cubic graph 𝐺 has a collection of at most five perfect matchings

with the property that each edge of 𝐺 is contained in at least one member of

them (or, equivalently, 𝒯5 = 𝒢).

The perfect matching covering in Conjecture 1.2 is called a Berge covering

of 𝐺.X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 5 / 29

Matching coverings

For cubic graphs,

𝒯 ⋆6 ⊆ 𝒯5.

However, it remains unknown whether 𝒯5 = 𝒯 ⋆6 . Under the assumption that

𝒯5 = 𝒢, Mazzuoccolo proved the following theorem.

Theorem 1.3 (Mazzuoccolo, 2011)

If 𝒯5 = 𝒢, then 𝒯5 = 𝒯 ⋆6 .

However, the equivalency of Berge’s Conjecture and Berge-Fulkerson

Conjecture remains unknown for a given graph.


Circuit/even subgraph covering

A cycle cover (or even subgraph cover) of a graph 𝐺 is a family ℱ of cycles

such that each edge of 𝐺 is contained by at least one member of ℱ .

Circuit double cover conjecture is one of major open problems in graph

theory. The following stronger version of the circuit double cover conjecture was

proposed by Celmins and Preissmann.

Conjecture 1.4 (Celmins, 1984 and Preissmann, 1981)

Every bridgeless graph 𝐺 has a 5-even subgraph double cover.

Note that, by applying Fleischner’s vertex splitting lemma, it suffices to

prove Conjecture 1.4 for cubic graphs only.


A family of Berge coverable graphs

We call a graph 𝐺 hypohamiltonian if 𝐺 itself is not hamiltonian but 𝐺− 𝑣

has a hamiltonian circuit for any vertex 𝑣 of 𝐺.

A snark is a non-3-edge-colorable cubic graph. Conjectures 1.1 and 1.2 are

trivial for 3-edge-colorable cubic graphs, especially for hamiltonian cubic graphs.

Haggkvist proposed a weak version of Conjecture 1.1.

Conjecture 2.1 (Haggkvist,2007)

Every hypohamiltonian cubic graph has a Fulkerson covering.


A family of Berge coverable graphs

A cubic graph 𝐺 is called a Kotzig graph if 𝐺 is 3-edge-colorable such that

each pair of colors form a hamiltonian circuit (defined by Haggkvist and

Markstrom, 2006). A cubic graph 𝐺 is called an almost Kotzig graph if, there is a

vertex 𝑤 of 𝐺, such that the suppressed graph 𝐺− 𝑤 is a Kotzig graph.

Theorem 2.2 (Hou, Lai, Zhang, 2012)

Let 𝐺 be an almost Kotzig graph. Then 𝐺 ∈ 𝒯5. That is, every almost Kotzig

graph has a Berge covering.

The result partially supports Haggkvist’s Conjecture.


A sufficient and necessary condition for a cubic graph

𝐺 admitting a Fulkerson Conjecture

Hao et al gave a sufficient and necessary condition for a given cubic graph

𝐺 ∈ 𝒯 ⋆6 . They proved the following lemma.

Lemma 2.3 (Hao et al, 2009)

Given a cubic graph 𝐺, 𝐺 ∈ 𝒯 ⋆6 if and only if there are two edge-disjoint

matchings 𝑀1 and 𝑀2 such that each suppressed graph 𝐺 ∖𝑀𝑖 is

3-edge-colorable for 𝑖 = 1, 2 and 𝑀1 ∪𝑀2 forms an even subgraph in 𝐺.


Outline of the proof of Theorem 2.2

We need only prove that 𝐺 admits a Fulkerson covering or a Berge covering.

Let 𝑛 = |𝑉 (𝐺)|. Then 𝑛 is even. Since 𝐺− 𝑤 is Kotzigian, 𝐺− 𝑤 has an

edge coloring 𝑓 : 𝐸(𝐺 ∖ 𝑤) → {1, 2, 3} such that each pair of colors form a

hamiltonian circuit of 𝐺− 𝑤. Let 𝑁𝐺(𝑤) = {𝑎, 𝑏, 𝑐}. For 𝑥 ∈ {𝑎, 𝑏, 𝑐}, let 𝑥1 and

𝑥2 denote the neighbors of 𝑥 other than 𝑤.



Case 1. 𝑎1𝑎2, 𝑏1𝑏2 and 𝑐1𝑐2 have the same color, say 3.

Let 𝑀𝑖 = {𝑒 | 𝑓(𝑒) = 𝑖}, 𝑖 = 1, 2. Then 𝑀1 and 𝑀2 are two matchings of

𝐺.

By the definition of 𝑓 , the edges of colors 1 and 2 form a hamiltonian

circuit, say 𝐶, of 𝐺− 𝑤. Then 𝐶 is also a circuit of 𝐺 consisting of two

matchings 𝑀1 and 𝑀2.

Extend 𝑓 to be an edge coloring (not proper) of 𝐺− 𝑤 such that 𝑥1𝑥 and

𝑥𝑥2 are colored by 𝑓(𝑥1𝑥2) for 𝑥 ∈ {𝑎, 𝑏, 𝑐} (see Fig.13).

By the definition of 𝑓 again, the edges of colors 𝑖 and 3 form a hamiltonian

circuit 𝐶𝑖 of 𝐺− 𝑤 for 𝑖 = 1, 2. Then 𝐶𝑖(𝑖 = 1, 2) is a circuit of 𝐺 of length

𝑛− 1. Since 𝑀𝑖 is the set of chords of 𝐶3−𝑖 for 𝑖 = 1, 2, 𝐺 ∖𝑀1∼= 𝐾4 is

3-edge-colorable. By Lemma 2.3, 𝐺 has a Fulkerson covering.



Case 2. 𝑎1𝑎2, 𝑏1𝑏2 and 𝑐1𝑐2 are assigned two colors.

Assume 𝑎1𝑎2 and 𝑏1𝑏2 have color 2 and 𝑐1𝑐2 has color 3(Fig. 14 (a)). For

𝑖 = 1, 2, 3, let 𝐸𝑖 = {𝑒 | 𝑓(𝑒) = 𝑖}.


Set

𝑀1 = {𝑤𝑎} ∪ (𝐸(𝐶𝑎𝑏) ∩ 𝐸3) ∪ (𝐸(𝐶𝑏𝑐 ∩ 𝐸2)) ∪ (𝐸(𝐶𝑐𝑎 ∩ 𝐸3))

and

𝑀2 = {𝑤𝑏} ∪ (𝐸(𝐶𝑎𝑏) ∩ 𝐸3) ∪ (𝐸(𝐶𝑏𝑐) ∩ 𝐸3) ∪ (𝐸(𝐶𝑐𝑎) ∩ 𝐸2).

Then 𝑀1 and 𝑀2 are two perfect matchings covering the edges 𝑤𝑎,𝑤𝑏 and

the edges of color 3 and part of edges of color 2.

Again by 𝐺− 𝑤 is Kotzigian, the circuit 𝐶 ′ of 𝐺 formed by the edges of

colors 1 and 2 has length 𝑛− 2 (see Fig. 14 (b)). Hence 𝐶 ′ can be partitioned

into two matchings 𝑀 ′3 and 𝑀 ′

4. Set 𝑀𝑖 = 𝑀 ′𝑖 ∪ {𝑤𝑐} for 𝑖 = 3, 4. Then 𝑀3 and

𝑀4 are two perfect matchings of 𝐺 covering the edges 𝑤𝑐 and the edges of colors

1 and 2.

Therefore {𝑀1,𝑀2,𝑀3,𝑀4} is a perfect matching covering (Berge

covering) of 𝐺.



Case 3. 𝑎1𝑎2, 𝑏1𝑏2 and 𝑐1𝑐2 have pairwise different colors.

Without loss of generality, assume 𝑎1𝑎2 has color 1, 𝑏1𝑏2 has color 2 and

𝑐1𝑐2 has color 3. Extend 𝑓 to be an edge coloring (not proper) of 𝐺− 𝑤 such

that 𝑥1𝑥 and 𝑥𝑥2 are colored by 𝑓(𝑥1𝑥2) for 𝑥 ∈ {𝑎, 𝑏, 𝑐}. Since 𝐺− 𝑤 is

Kotzigian, the circuit 𝐶𝑖𝑗 of 𝐺 formed by the edges of colors 𝑖 and 𝑗 has length

𝑛− 2 for 𝑖, 𝑗 ∈ {1, 2, 3} (𝐶12 is as shown in Fig.??). Hence 𝐶𝑖𝑗 can be

partitioned into two matchings, say 𝑀 ′𝑖𝑗 and 𝑀 ′′

𝑖𝑗 , for 𝑖, 𝑗 ∈ {1, 2, 3}. Set𝑀1 = 𝑀 ′

12 ∪ {𝑤𝑐}, 𝑀2 = 𝑀 ′′12 ∪ {𝑤𝑐}, 𝑀3 = 𝑀 ′

13 ∪ {𝑤𝑏}, 𝑀4 = 𝑀 ′′13 ∪ {𝑤𝑏},

𝑀5 = 𝑀 ′23 ∪ {𝑤𝑎}, and 𝑀6 = 𝑀 ′′

23 ∪ {𝑤𝑎}. It is an easy task to check that

{𝑀1,𝑀2, · · · ,𝑀6} is a Fulkerson covering of 𝐺.


Matching coverings and cycle coverings

𝒯4-graphs (the family of cubic graphs that are covered by a set of four

perfect matchings) are somehow special, and are expected to have some special

graph theory properties.

Conjecture 1.4 has been verified by Huck and Kochol [2] for oddness 2

graphs. Here, we will verify Conjecture 1.4 for graphs in 𝒯4.

Theorem 3.1

If 𝐺 ∈ 𝒯4, then 𝐺 has a 5-even subgraph double cover.

Theorem 3.1 was also proved by Steffen [3] independently.


Matching coverings and cycle coverings

Here, we present a stronger version.


Let 𝐺 be a cubic graph. The following statements are equivalent.

(1) 𝐺 ∈ 𝒯4;(2) 𝐺 has a 5-even subgraph double cover {𝐶0, . . . , 𝐶4} with 𝐶0 as a 2-factor.

Outline of the proof: (1) ⇒ (2): Let ℳ = {𝑀1, . . . ,𝑀4} be a perfect

matching covering of 𝐺. Denote

𝐸𝜇 = {𝑒 ∈ 𝐸(𝐺) : 𝑒 is covered by ℳ 𝜇-times}.

Hence 𝐸1 = △4𝑖=1𝑀𝑖 is a 2-factor. And 𝑀 𝑐

𝑖 = 𝐺− 𝐸(𝑀𝑖) is also a 2-factor

(𝑖 = 1, 2, 3, 4).


Outline of the proof

Thus

{𝐸1 △𝑀 𝑐𝑖 : 𝑖 = 1, . . . , 4} ∪ {𝐸1}

is a 5-even subgraph double cover of 𝐺.

(2) ⇒ (1): We claim that {𝐶0 △ 𝐶𝑖 : 𝑖 = 1, 2, 3, 4} is a set of 2-factors.

Thus, we can see that {𝐶0 △ 𝐶𝑖 : 𝑖 = 1, 2, 3, 4} covers every edge twice or

three times.

So, 𝑀𝑖 = 𝐸(𝐺)− 𝐸(𝐶0 △ 𝐶𝑖) is a perfect matching, and {𝑀1, . . . ,𝑀4}covers each edge 𝑒 once if 𝑒 ∈ 𝐸(𝐶0), or, twice if 𝑒 /∈ 𝐸(𝐶0).


Parity subgraph covering

Let 𝐺 be a graph. A subgraph 𝑃 of 𝐺 is a parity subgraph of 𝐺 if

𝑑𝑃 (𝑣) ≡ 𝑑𝐺(𝑣) (mod 2) for every vertex 𝑣 of 𝐺. It is evident that 𝑃 is a parity

subgraph if and only if 𝐺− 𝐸(𝑃 ) is even.

Note that, graphs considered here may not be necessary cubic.

A set 𝒫 of parity subgraphs of 𝐺 is a (1, 2)-covering if every edge of 𝐺 is

contained in precisely one or two members of 𝒫. Let 𝒮⋆𝜇 be the set of graphs

admitting parity subgraph (1, 2)-coverings 𝒫 with |𝒫| = 𝜇.


Parity subgraph covering and cycle covering

Clearly, 𝒯4 ⊆ 𝒮*4 . The following is an analogy of Theorem 3.2 for parity

subgraph covering, and is another stronger version of Theorem 3.1. It is also an

equivalent version for the 5-even subgraph double cover problem.


Let 𝐺 be a graph. The following statements are equivalent.

(1) 𝐺 ∈ 𝒮⋆4 ;

(2) 𝐺 has a 5-even subgraph double cover.

Outline of the proof: The proof is similar to Theorem 3.2.


Matching coverings

Although every cubic graph is conjectured to have a 5-even subgraph double

cover (Conjecture 1.4), not every graph has the 𝒯4-property. The Petersen graph

is an example. The Petersen graph has precisely six perfect matchings, and each

pair of them intersect at precisely one edge. Thus, we have the following

proposition.

Proposition 3.4 ((Fouquet and Vanherpe, 2009))

The Petersen graph 𝑃10 is not a member of 𝒯4.


Matching coverings: Conjectures

We propose the following conjectures.

Conjecture 3.5

For every bridgeless cubic graph 𝐺, if 𝐺 is Petersen-minor-free, then 𝐺 ∈ 𝒯4.

This is a weak version of a conjecture by Tutte that 𝐺 ∈ 𝒯3 for every

bridgeless Petersen minor-free cubic graph 𝐺. Although the proof of this Tutte’s

conjecture was announced by Robertson, Sanders, Seymour and Thomas ([5] [2]),

a simplified manual proof of Conjecture 3.5 will certainly develop some new

techniques in graph theory, and, therefore, it remains as an interesting research

problem in graph theory.



Problem 3.6 (Fouquet and Vanherpe, 2009)

Except for the Petersen graph, is there any cyclically 4-edge-connected cubic

graph 𝐺 that is not a 𝒯4-graph?

The condition cyclically 4-edge connected in Problem 3.6 is tight, Fouquet

and Vanherpe [3] gave two families of 3-edge-connected non-𝒯4-graphs.By Theorem 3.2, Problem 3.6 implies Conjecture 1.4. Problem 3.6 has been

supported by a recent result ([3]) that every permutation graph is a 𝒯4-graphexcept for the Petersen graph.



Conjecture 3.7

For every given cubic graph 𝐺, if 𝐺 ∈ 𝒯4 then 𝐺 ∈ 𝒯 ⋆6 . That is, Berge-Fulkerson

conjecture is true for all graphs of 𝒯4.

Conjecture 3.8

For every given snark 𝐺, 𝐺 ∈ 𝒯5 if and only if 𝐺 ∈ 𝒯 ⋆6 .

Conjecture 3.8 implies the equivalence of Conjecture 1.1 and Conjecture 1.2

for every given graph (a further improvement of Theorem 1.3). Some families of

cubic graphs have been confirmed as 𝒯5-graphs (such as, permutation graphs ([3],

Theorem 3.12), almost Kotzig graphs (Theorem 2.2), etc.). The verification of

Conjecture 3.8 will further extend those results for Berge-Fulkerson conjecture.


References

U.A. Celmins, On cubic graphs that do not have an edge 3-coloring. Ph.D.

thesis, University of Waterloo, Ontario, Canada. (1984)

H. Fleischner, H., Eine gemeinsame Basis fur die Theorie der eulerschen

Graphen und den Satz von Petersen, Monatsh. Math., 81 (1976) 267-278.

J. L. Fouquet and J. M. Vanherpe, On the perfect matching index of

bridgeless cubic graphs, arXiv:0904.1296.

D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math.

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R. Hao, J. Niu, X. Wang, C.-Q. Zhang, T. Zhang, A note on

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650-658.

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References

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References

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THANK YOU VERY MUCH!


on matching coverings and cycle coverings · 2013-09-29 · circuit double cover conjecture is one...

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