On matching coverings and cycle coverings Xinmin Hou (co-work with - - PowerPoint PPT Presentation
On matching coverings and cycle coverings Xinmin Hou (co-work with Hong-Jian Lai and Cun-Quan Zhang) Email: xmhou@ustc.edu.cn School of of Mathematical Science University of Science and Technology of China Hefei, Anhui 230026, China X. Hou,
Xinmin Hou (co-work with Hong-Jian Lai and Cun-Quan Zhang) Email: xmhou@ustc.edu.cn
School of of Mathematical Science University of Science and Technology of China Hefei, Anhui 230026, China
On matching coverings and cycle coverings 2012-10 1 / 29
1
Defjnitions and Conjectures
2
A family of Berge coverable graphs
3
Matching coverings and cycle coverings
On matching coverings and cycle coverings 2012-10 2 / 29
1
Defjnitions and Conjectures
2
A family of Berge coverable graphs
3
Matching coverings and cycle coverings
On matching coverings and cycle coverings 2012-10 3 / 29
Let π» be a graph. A matching π is a 1-regular subgraph of π». A perfect matching of π» is a spanning 1-regular subgraph of π» (also called a 1-factor of π»), and an π -factor of π» is a spanning π -regular subgraph of π».
On matching coverings and cycle coverings 2012-10 3 / 29
Let π» be a graph. A matching π is a 1-regular subgraph of π». A perfect matching of π» is a spanning 1-regular subgraph of π» (also called a 1-factor of π»), and an π -factor of π» is a spanning π -regular subgraph of π». 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.
On matching coverings and cycle coverings 2012-10 3 / 29
Let π» be a graph. A matching π is a 1-regular subgraph of π». A perfect matching of π» is a spanning 1-regular subgraph of π» (also called a 1-factor of π»), and an π -factor of π» is a spanning π -regular subgraph of π». 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 π», is the graph obtained from π» by suppressing all degree two vertices.
On matching coverings and cycle coverings 2012-10 3 / 29
A perfect matching covering β³ of π» is a set of perfect matchings of π» if every edge of π» is contained in at least one member of β³. Let π°π be the set of cubic graphs admitting perfect matching coverings β³ with |β³| = π.
On matching coverings and cycle coverings 2012-10 4 / 29
A perfect matching covering β³ of π» is a set of perfect matchings of π» if every edge of π» is contained in at least one member of β³. Let π°π be the set of cubic graphs admitting perfect matching coverings β³ with |β³| = π. A perfect matching covering β³ of π» is a (1, 2)-covering if every edge of π» is contained in precisely one or two members of β³. Let π° β
π be the set of cubic
graphs admitting perfect matching (1, 2)-coverings β³ with |β³| = π.
On matching coverings and cycle coverings 2012-10 4 / 29
Let π£ be the family of all bridgeless cubic graphs.
Conjecture 1.1 (Berge-Fulkerson Conjecture)
Every bridgeless cubic graph π» has a collection of six perfect matchings that together cover every edge of π» exactly twice (or, equivalently, π° β
6 = π£).
We call such a perfect matching covering in Conjecture 1.1 a Fulkerson covering.
On matching coverings and cycle coverings 2012-10 5 / 29
Let π£ be the family of all bridgeless cubic graphs.
Conjecture 1.1 (Berge-Fulkerson Conjecture)
Every bridgeless cubic graph π» has a collection of six perfect matchings that together cover every edge of π» exactly twice (or, equivalently, π° β
6 = π£).
We call such a perfect matching covering in Conjecture 1.1 a Fulkerson covering.
Conjecture 1.2 (Bergeβs Conjecture)
Every bridgeless cubic graph π» has a collection of at most fjve perfect matchings with the property that each edge of π» is contained in at least one member of them (or, equivalently, π°5 = π£). The perfect matching covering in Conjecture 1.2 is called a Berge covering
On matching coverings and cycle coverings 2012-10 5 / 29
For cubic graphs, π° β
6 β π°5.
However, it remains unknown whether π°5 = π° β
6 . Under the assumption that
π°5 = π£, Mazzuoccolo proved the following theorem.
Theorem 1.3 (Mazzuoccolo, 2011)
If π°5 = π£, then π°5 = π° β
6 .
However, the equivalency of Bergeβs Conjecture and Berge-Fulkerson Conjecture remains unknown for a given graph.
On matching coverings and cycle coverings 2012-10 6 / 29
A cycle cover (or even subgraph cover) of a graph π» is a family β± of cycles such that each edge of π» is contained by at least one member of β±.
On matching coverings and cycle coverings 2012-10 7 / 29
A cycle cover (or even subgraph cover) of a graph π» is a family β± of cycles such that each edge of π» is contained by at least one member of β±. Circuit double cover conjecture is one of major open problems in graph
proposed by Celmins and Preissmann.
Conjecture 1.4 (Celmins, 1984 and Preissmann, 1981)
Every bridgeless graph π» has a 5-even subgraph double cover. Note that, by applying Fleischnerβs vertex splitting lemma, it suffjces to prove Conjecture 1.4 for cubic graphs only.
On matching coverings and cycle coverings 2012-10 7 / 29
1
Defjnitions and Conjectures
2
A family of Berge coverable graphs
3
Matching coverings and cycle coverings
On matching coverings and cycle coverings 2012-10 8 / 29
We call a graph π» hypohamiltonian if π» itself is not hamiltonian but π» β π€ has a hamiltonian circuit for any vertex π€ of π».
On matching coverings and cycle coverings 2012-10 8 / 29
We call a graph π» hypohamiltonian if π» itself is not hamiltonian but π» β π€ has a hamiltonian circuit for any vertex π€ of π». 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.
On matching coverings and cycle coverings 2012-10 8 / 29
We call a graph π» hypohamiltonian if π» itself is not hamiltonian but π» β π€ has a hamiltonian circuit for any vertex π€ of π». 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. HΒ¨ aggkvist proposed a weak version of Conjecture 1.1.
Conjecture 2.1 (H¨ aggkvist,2007)
Every hypohamiltonian cubic graph has a Fulkerson covering.
On matching coverings and cycle coverings 2012-10 8 / 29
A cubic graph π» is called a Kotzig graph if π» is 3-edge-colorable such that each pair of colors form a hamiltonian circuit (defjned by HΒ¨ aggkvist and MarkstrΒ¨
vertex π₯ of π», such that the suppressed graph π» β π₯ is a Kotzig graph.
On matching coverings and cycle coverings 2012-10 9 / 29
A cubic graph π» is called a Kotzig graph if π» is 3-edge-colorable such that each pair of colors form a hamiltonian circuit (defjned by HΒ¨ aggkvist and MarkstrΒ¨
vertex π₯ of π», such that the suppressed graph π» β π₯ is a Kotzig graph.
Theorem 2.2 (Hou, Lai, Zhang, 2012)
Let π» be an almost Kotzig graph. Then π» β π°5. That is, every almost Kotzig graph has a Berge covering. The result partially supports HΒ¨ aggkvistβs Conjecture.
On matching coverings and cycle coverings 2012-10 9 / 29
Hao et al gave a suffjcient and necessary condition for a given cubic graph π» β π° β
6 . They proved the following lemma.
Lemma 2.3 (Hao et al, 2009)
Given a cubic graph π», π» β π° β
6 if and only if there are two edge-disjoint
matchings π1 and π2 such that each suppressed graph π» β ππ is 3-edge-colorable for π = 1, 2 and π1 βͺ π2 forms an even subgraph in π».
On matching coverings and cycle coverings 2012-10 10 / 29
We need only prove that π» admits a Fulkerson covering or a Berge covering. Let π = |π (π»)|. Then π is even. Since π» β π₯ is Kotzigian, π» β π₯ has an edge coloring π : πΉ(π» β π₯) β {1, 2, 3} such that each pair of colors form a hamiltonian circuit of π» β π₯. Let ππ»(π₯) = {π, π, π}. For π¦ β {π, π, π}, let π¦1 and π¦2 denote the neighbors of π¦ other than π₯.
On matching coverings and cycle coverings 2012-10 11 / 29
Case 1. π1π2, π1π2 and π1π2 have the same color, say 3. Let ππ = {π | π(π) = π}, π = 1, 2. Then π1 and π2 are two matchings of π». By the defjnition of π, the edges of colors 1 and 2 form a hamiltonian circuit, say π·, of π» β π₯. Then π· is also a circuit of π» consisting of two matchings π1 and π2. Extend π to be an edge coloring (not proper) of π» β π₯ such that π¦1π¦ and π¦π¦2 are colored by π(π¦1π¦2) for π¦ β {π, π, π} (see Fig.13). By the defjnition of π again, the edges of colors π and 3 form a hamiltonian circuit π·π of π» β π₯ for π = 1, 2. Then π·π(π = 1, 2) is a circuit of π» of length π β 1. Since ππ is the set of chords of π·3βπ for π = 1, 2, π» β π1 βΌ = πΏ4 is 3-edge-colorable. By Lemma 2.3, π» has a Fulkerson covering.
On matching coverings and cycle coverings 2012-10 12 / 29
On matching coverings and cycle coverings 2012-10 13 / 29
Case 2. π1π2, π1π2 and π1π2 are assigned two colors. Assume π1π2 and π1π2 have color 2 and π1π2 has color 3(Fig. 14 (a)). For π = 1, 2, 3, let πΉπ = {π | π(π) = π}.
On matching coverings and cycle coverings 2012-10 14 / 29
Set π1 = {π₯π} βͺ (πΉ(π·ππ) β© πΉ3) βͺ (πΉ(π·ππ β© πΉ2)) βͺ (πΉ(π·ππ β© πΉ3)) and π2 = {π₯π} βͺ (πΉ(π·ππ) β© πΉ3) βͺ (πΉ(π·ππ) β© πΉ3) βͺ (πΉ(π·ππ) β© πΉ2). Then π1 and π2 are two perfect matchings covering the edges π₯π, π₯π and the edges of color 3 and part of edges of color 2. Again by π» β π₯ is Kotzigian, the circuit π·β² of π» formed by the edges of colors 1 and 2 has length π β 2 (see Fig. 14 (b)). Hence π·β² can be partitioned into two matchings π β²
3 and π β²
π βͺ {π₯π} for π = 3, 4. Then π3 and
π4 are two perfect matchings of π» covering the edges π₯π and the edges of colors 1 and 2. Therefore {π1, π2, π3, π4} is a perfect matching covering (Berge covering) of π».
On matching coverings and cycle coverings 2012-10 15 / 29
Case 3. π1π2, π1π2 and π1π2 have pairwise difgerent colors.
On matching coverings and cycle coverings 2012-10 16 / 29
1
Defjnitions and Conjectures
2
A family of Berge coverable graphs
3
Matching coverings and cycle coverings
On matching coverings and cycle coverings 2012-10 17 / 29
π°4-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.
On matching coverings and cycle coverings 2012-10 17 / 29
π°4-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 verifjed by Huck and Kochol [2] for oddness 2
Theorem 3.1
If π» β π°4, then π» has a 5-even subgraph double cover. Theorem 3.1 was also proved by Stefgen [3] independently.
On matching coverings and cycle coverings 2012-10 17 / 29
Here, we present a stronger version.
Theorem 3.2 (Hou, Lai, Zhang, 2012)
Let π» be a cubic graph. The following statements are equivalent. (1) π» β π°4; (2) π» has a 5-even subgraph double cover {π·0, . . . , π·4} with π·0 as a 2-factor. Outline of the proof: (1) β (2): Let β³ = {π1, . . . , π4} be a perfect matching covering of π». Denote πΉπ = {π β πΉ(π») : π is covered by β³ π-times}. Hence πΉ1 = β³4
π=1ππ is a 2-factor. And π π π = π» β πΉ(ππ) is also a 2-factor
(π = 1, 2, 3, 4).
On matching coverings and cycle coverings 2012-10 18 / 29
Thus {πΉ1 β³ π π
π : π = 1, . . . , 4} βͺ {πΉ1}
is a 5-even subgraph double cover of π». (2) β (1): We claim that {π·0 β³ π·π : π = 1, 2, 3, 4} is a set of 2-factors. Thus, we can see that {π·0 β³ π·π : π = 1, 2, 3, 4} covers every edge twice or three times. So, ππ = πΉ(π») β πΉ(π·0 β³ π·π) is a perfect matching, and {π1, . . . , π4} covers each edge π once if π β πΉ(π·0), or, twice if π / β πΉ(π·0).
On matching coverings and cycle coverings 2012-10 19 / 29
Let π» be a graph. A subgraph π of π» is a parity subgraph of π» if ππ (π€) β‘ ππ»(π€) (mod 2) for every vertex π€ of π». It is evident that π is a parity subgraph if and only if π» β πΉ(π) is even. Note that, graphs considered here may not be necessary cubic.
On matching coverings and cycle coverings 2012-10 20 / 29
Let π» be a graph. A subgraph π of π» is a parity subgraph of π» if ππ (π€) β‘ ππ»(π€) (mod 2) for every vertex π€ of π». It is evident that π is a parity subgraph if and only if π» β πΉ(π) is even. Note that, graphs considered here may not be necessary cubic. A set π¬ of parity subgraphs of π» is a (1, 2)-covering if every edge of π» is contained in precisely one or two members of π¬. Let π―β
π be the set of graphs
admitting parity subgraph (1, 2)-coverings π¬ with |π¬| = π.
On matching coverings and cycle coverings 2012-10 20 / 29
Clearly, π°4 β π―*
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 π» be a graph. The following statements are equivalent. (1) π» β π―β
4;
(2) π» has a 5-even subgraph double cover. Outline of the proof: The proof is similar to Theorem 3.2.
On matching coverings and cycle coverings 2012-10 21 / 29
Although every cubic graph is conjectured to have a 5-even subgraph double cover (Conjecture 1.4), not every graph has the π°4-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 π10 is not a member of π°4.
On matching coverings and cycle coverings 2012-10 22 / 29
We propose the following conjectures.
Conjecture 3.5
For every bridgeless cubic graph π», if π» is Petersen-minor-free, then π» β π°4. This is a weak version of a conjecture by Tutte that π» β π°3 for every bridgeless Petersen minor-free cubic graph π». Although the proof of this Tutteβs conjecture was announced by Robertson, Sanders, Seymour and Thomas ([5] [2]), a simplifjed 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.
On matching coverings and cycle coverings 2012-10 23 / 29
Problem 3.6 (Fouquet and Vanherpe, 2009)
Except for the Petersen graph, is there any cyclically 4-edge-connected cubic graph π» that is not a π°4-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-π°4-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 π°4-graph except for the Petersen graph.
On matching coverings and cycle coverings 2012-10 24 / 29
Conjecture 3.7
For every given cubic graph π», if π» β π°4 then π» β π° β
6 . That is, Berge-Fulkerson
conjecture is true for all graphs of π°4.
Conjecture 3.8
For every given snark π», π» β π°5 if and only if π» β π° β
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 confjrmed as π°5-graphs (such as, permutation graphs ([3], Theorem 3.12), almost Kotzig graphs (Theorem 2.2), etc.). The verifjcation of Conjecture 3.8 will further extend those results for Berge-Fulkerson conjecture.
On matching coverings and cycle coverings 2012-10 25 / 29
U.A. Celmins, On cubic graphs that do not have an edge 3-coloring. Ph.D. thesis, University of Waterloo, Ontario, Canada. (1984)
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aggkvist, Problem 443. Special case of the Fulkerson Conjecture, in B. Mohar, R. J. Nowakowski, D. B. West, βResearch problems from the 5th Slovenian Conference(Bled, 2003)β, Discrete Mathematics 307 (2007) (3-5): 650-658.
On matching coverings and cycle coverings 2012-10 26 / 29
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On matching coverings and cycle coverings 2012-10 27 / 29
U.S.R. (eds), Graph Theory and Related Topics. New York: Academic Press. (1979)
talk at Graph Theory at Georgia Tech, May 10th, Atlanta, 2012.
http://people.math.gatech.edu/thomas/FC/generalize.html. C.-Q. Zhang, Circuit Double Covers of Graphs, Cambridge University Press, (2012) ISBN: 978-0-521-28235-2.
On matching coverings and cycle coverings 2012-10 28 / 29
On matching coverings and cycle coverings 2012-10 29 / 29