on matching coverings and cycle coverings · 2013-09-29 · circuit double cover conjecture is one...
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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
Contents
1 Definitions and Conjectures
2 A family of Berge coverable graphs
3 Matching coverings and cycle coverings
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 2 / 29
1 Definitions and Conjectures
2 A family of Berge coverable graphs
3 Matching coverings and cycle coverings
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 3 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 3 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 3 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 3 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 4 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 4 / 29
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
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 6 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 7 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 7 / 29
1 Definitions and Conjectures
2 A family of Berge coverable graphs
3 Matching coverings and cycle coverings
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 8 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 8 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 8 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 8 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 9 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 9 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 10 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 11 / 29
Outline of the proof of Theorem 2.2
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 12 / 29
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 13 / 29
Outline of the proof of Theorem 2.2
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}.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 14 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 15 / 29
Outline of the proof of Theorem 2.2
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 16 / 29
1 Definitions and Conjectures
2 A family of Berge coverable graphs
3 Matching coverings and cycle coverings
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 17 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 17 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 17 / 29
Matching coverings and cycle coverings
Here, we present a stronger version.
Theorem 3.2 (Hou, Lai, Zhang, 2012)
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).
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 18 / 29
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).
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 19 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 20 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 20 / 29
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.
Theorem 3.3 (Hou, Lai, Zhang, 2012)
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 21 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 22 / 29
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 23 / 29
Matching coverings: Conjectures
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 24 / 29
Matching coverings: Conjectures
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.
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 25 / 29
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X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 28 / 29
THANK YOU VERY MUCH!
X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 29 / 29