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

Contents 1 Defjnitions and Conjectures A family of Berge coverable graphs 2 Matching coverings and cycle coverings 3 X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 2 / 29

Defjnitions and Conjectures 1 A family of Berge coverable graphs 2 Matching coverings and cycle coverings 3 X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 3 / 29

Defjnitions Let 𝐻 be a graph. A matching 𝑁 is a 1 -regular subgraph of 𝐻 . A perfect matching of 𝐻 is a spanning 1 -regular subgraph of 𝐻 (also called a 1 -factor of 𝐻 ), and an 𝑠 -factor of 𝐻 is a spanning 𝑠 -regular subgraph of 𝐻 . X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 3 / 29

Defjnitions Let 𝐻 be a graph. A matching 𝑁 is a 1 -regular subgraph of 𝐻 . A perfect matching of 𝐻 is a spanning 1 -regular subgraph of 𝐻 (also called a 1 -factor of 𝐻 ), and an 𝑠 -factor of 𝐻 is a spanning 𝑠 -regular subgraph of 𝐻 . 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. X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 3 / 29

Defjnitions Let 𝐻 be a graph. A matching 𝑁 is a 1 -regular subgraph of 𝐻 . A perfect matching of 𝐻 is a spanning 1 -regular subgraph of 𝐻 (also called a 1 -factor of 𝐻 ), and an 𝑠 -factor of 𝐻 is a spanning 𝑠 -regular subgraph of 𝐻 . 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 𝐻 , is the graph obtained from 𝐻 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 𝐻 is a set of perfect matchings of 𝐻 if every edge of 𝐻 is contained in at least one member of ℳ . Let 𝒰 𝜈 be the set of cubic graphs admitting perfect matching coverings ℳ with |ℳ| = 𝜈 . X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 4 / 29

Matching coverings A perfect matching covering ℳ of 𝐻 is a set of perfect matchings of 𝐻 if every edge of 𝐻 is contained in at least one member of ℳ . Let 𝒰 𝜈 be the set of cubic graphs admitting perfect matching coverings ℳ with |ℳ| = 𝜈 . A perfect matching covering ℳ of 𝐻 is a (1 , 2) -covering if every edge of 𝐻 is contained in precisely one or two members of ℳ . Let 𝒰 ⋆ 𝜈 be the set of cubic graphs admitting perfect matching (1 , 2) -coverings ℳ with |ℳ| = 𝜈 . X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 4 / 29

Matching coverings Let 𝒣 be the family of all bridgeless cubic graphs. Conjecture 1.1 (Berge-Fulkerson Conjecture) Every bridgeless cubic graph 𝐻 has a collection of six perfect matchings that together cover every edge of 𝐻 exactly twice (or, equivalently, 𝒰 ⋆ 6 = 𝒣 ). We call such a perfect matching covering in Conjecture 1.1 a Fulkerson covering . X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 5 / 29

Matching coverings Let 𝒣 be the family of all bridgeless cubic graphs. Conjecture 1.1 (Berge-Fulkerson Conjecture) Every bridgeless cubic graph 𝐻 has a collection of six perfect matchings that together cover every edge of 𝐻 exactly twice (or, equivalently, 𝒰 ⋆ 6 = 𝒣 ). We call such a perfect matching covering in Conjecture 1.1 a Fulkerson covering . Conjecture 1.2 (Berge’s Conjecture) Every bridgeless cubic graph 𝐻 has a collection of at most fjve perfect matchings with the property that each edge of 𝐻 is contained in at least one member of them (or, equivalently, 𝒰 5 = 𝒣 ). The perfect matching covering in Conjecture 1.2 is called a Berge covering of 𝐻 . X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 5 / 29

Matching coverings For cubic graphs, 𝒰 ⋆ 6 ⊆ 𝒰 5 . However, it remains unknown whether 𝒰 5 = 𝒰 ⋆ 6 . Under the assumption that 𝒰 5 = 𝒣 , Mazzuoccolo proved the following theorem. Theorem 1.3 (Mazzuoccolo, 2011) If 𝒰 5 = 𝒣 , then 𝒰 5 = 𝒰 ⋆ 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 𝐻 is a family ℱ of cycles such that each edge of 𝐻 is contained by at least one member of ℱ . 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 𝐻 is a family ℱ of cycles such that each edge of 𝐻 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 𝐻 has a 5 -even subgraph double cover. Note that, by applying Fleischner’s vertex splitting lemma, it suffjces 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

Defjnitions and Conjectures 1 A family of Berge coverable graphs 2 Matching coverings and cycle coverings 3 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 𝐻 hypohamiltonian if 𝐻 itself is not hamiltonian but 𝐻 − 𝑤 has a hamiltonian circuit for any vertex 𝑤 of 𝐻 . 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 𝐻 hypohamiltonian if 𝐻 itself is not hamiltonian but 𝐻 − 𝑤 has a hamiltonian circuit for any vertex 𝑤 of 𝐻 . 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. 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 𝐻 hypohamiltonian if 𝐻 itself is not hamiltonian but 𝐻 − 𝑤 has a hamiltonian circuit for any vertex 𝑤 of 𝐻 . 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. H¨ aggkvist proposed a weak version of Conjecture 1.1. Conjecture 2.1 (H¨ aggkvist,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 𝐻 is called a Kotzig graph if 𝐻 is 3-edge-colorable such that each pair of colors form a hamiltonian circuit (defjned by H¨ aggkvist and Markstr¨ om, 2006). A cubic graph 𝐻 is called an almost Kotzig graph if, there is a vertex 𝑥 of 𝐻 , such that the suppressed graph 𝐻 − 𝑥 is a Kotzig graph. 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 𝐻 is called a Kotzig graph if 𝐻 is 3-edge-colorable such that each pair of colors form a hamiltonian circuit (defjned by H¨ aggkvist and Markstr¨ om, 2006). A cubic graph 𝐻 is called an almost Kotzig graph if, there is a vertex 𝑥 of 𝐻 , such that the suppressed graph 𝐻 − 𝑥 is a Kotzig graph. Theorem 2.2 (Hou, Lai, Zhang, 2012) Let 𝐻 be an almost Kotzig graph. Then 𝐻 ∈ 𝒰 5 . That is, every almost Kotzig graph has a Berge covering. The result partially supports H¨ aggkvist’s Conjecture. X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 9 / 29

A suffjcient and necessary condition for a cubic graph 𝐻 admitting a Fulkerson Conjecture Hao et al gave a suffjcient and necessary condition for a given cubic graph 𝐻 ∈ 𝒰 ⋆ 6 . They proved the following lemma. Lemma 2.3 (Hao et al , 2009) Given a cubic graph 𝐻 , 𝐻 ∈ 𝒰 ⋆ 6 if and only if there are two edge-disjoint matchings 𝑁 1 and 𝑁 2 such that each suppressed graph 𝐻 ∖ 𝑁 𝑗 is 3-edge-colorable for 𝑗 = 1 , 2 and 𝑁 1 ∪ 𝑁 2 forms an even subgraph in 𝐻 . 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 𝐻 admits a Fulkerson covering or a Berge covering. Let 𝑜 = | 𝑊 ( 𝐻 ) | . Then 𝑜 is even. Since 𝐻 − 𝑥 is Kotzigian, 𝐻 − 𝑥 has an edge coloring 𝑔 : 𝐹 ( 𝐻 ∖ 𝑥 ) → { 1 , 2 , 3 } such that each pair of colors form a hamiltonian circuit of 𝐻 − 𝑥 . Let 𝑂 𝐻 ( 𝑥 ) = { 𝑏, 𝑐, 𝑑 } . For 𝑦 ∈ { 𝑏, 𝑐, 𝑑 } , let 𝑦 1 and 𝑦 2 denote the neighbors of 𝑦 other than 𝑥 . X. Hou, C.-Q. Zhang (USTC) On matching coverings and cycle coverings 2012-10 11 / 29