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(1)國立交通大學應用數學系碩士論文. 圖的圈覆蓋. Cycle Cover of Graphs. 研究生：連敏筠指導教授：傅恆霖. 教授. 中華民國九十七年一月.

(2) 圖的圈覆蓋 Cycle Cover of Graphs 研究生：連敏筠. Student：Min-Yun Lien. 指導教授：傅恆霖. Advisor：Hung-Lin Fu. 國立交通大學應用數學系碩士論文. A Thesis Submitted to Department of Applied Mathematics College of Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in Applied Mathematics Jan 2008 Hsinchu, Taiwan, Republic of China. 中華民國九十七年一月.

(3) Cycle Cover of Graphs Student: Min-Yun Lien. Advisor: Hung-Lin Fu. Department of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan 30050.

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(11) Cycle Cover of Graphs Student:Min-Yun Lien Advisor:Hung-Lin Fu Department of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan 30050. Abstract A cycle cover of a graph G is a collection of cycles of G which covers all edges of G. The size of a cycle cover is the sum of the lengths of the cycles in the cover. A flow in G under orientation D is an integer-valued function φ on E(G) such that the output P P value e∈E + (v) φ(e) is equal to the input value e∈E − (v) φ(e) for each v ∈ V (G). The support of φ is defined by S(φ) = {e ∈ E(G) : φ(e) 6= 0}. For a positive integer k, if −k < φ(e) < k for every e ∈ E(G), then φ is called a k-flow, and furthermore, if S(φ) = E(G), then φ is called a nowhere-zero k-flow. In this thesis we prove: (1) if Tutte’s 3-Flow Conjecture is true, then every (k − 1)-edge-connected graph G with δ(G) = k has a nowhere-zero 6-flow φ such that when k is odd |Eodd (φ)|≥ and when k is even |Eodd (φ)|≥. k−2 |E(G)| k−1. ; (2) If a (k−1)-edge-connected graph G with. δ(G) = k has a nowhere-zero 6-flow φ such that when k is odd |Eodd (φ)|≥ then G has a cycle cover in which the size of the cycle cover is at most. k−1 |E(G)|, k. 13k+5 |E(G)| 9k. k−2 |E(G)|, k−1. then G has a cycle cover in which the size. 13k−8 |E(G)|, 9(k−1). where Eodd (φ)={e ∈ E(G) : φ(e) is odd }.. and when k is even |Eodd (φ)|≥ of the cycle cover is at most. k−1 |E(G)| k. ii.

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(13) Contents Abstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. i. Abstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ii. Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iii. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iv. 1 Introduction and Preliminaries. 1. 1.1. Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2 Known Results and Conjectures. 9. 2.1. Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.2. Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 3 Main Results and Conclusion. 14. 3.1. The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 3.2. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. iv.

(14) Chapter 1 Introduction and Preliminaries ”Cycle cover” has been one of the most important topics in Graph Theory. A cycle cover of a graph G is a collection of cycles of G which covers all edges of G. Since a graph is even if and only if it has a decomposition into edge-disjoint cycles, we also regard a cycle cover as a collection of even subgraphs. The size of a cycle cover is the sum of the lengths of the cycles in the cover. The problem of finding a cycle cover of small size (a short cycle cover) has been studied by several authors[3, 12, 21]. The best known result on this subject is the one by Bermond, Jackson and Jaeger [3], and independently by Alon and Tarsi [1], that every bridgeless graph G has a cycle cover of size at most 35 |E(G)|. Later, this upper bound was improved to. 44 |E(G)| 27. by. Fan in [8] under the assumption that Tutte’s 3-flow conjecture is true. The focus of this study will be on the graphs G with larger edge-connectivity. In which, under the same assumption, we are able to obtain a smaller upper bound with respect to the ratio of |E(G)| and the size of cycle cover.. 1.1. Basic Notations. A graph G is composed of two types of objects. It has a finite set of elements called vertices and a set of unordered pairs of vertices called edges. The vertex set is denoted by V (G) or V , and the edge set is denoted by E(G) or E. 1.

(15) A directed graph or digraph G is composed of two types of objects. It has a vertex set V (G) and an edge set E(G), and the edge set is a set of ordered pairs of vertices. The first vertex of the ordered pair is the tail of the edge, and the second is the head. We say that an edge in a digraph is an edge from its tail to its head. A loop is an edge whose endvertices are equal. Multiple edges are edges having the same pair of endvertices. A simple graph is a graph having no loops or multiple edges. A subgraph of a graph G is a graph H such that V (H) ⊆ V (G) and E(H) ⊆ E(G). A spanning subgraph of a graph G is a subgraph with vertex set V (G). The components of a graph G are its maximal connected subgraphs. The degree of vertex v in a graph G is the number of edges incident to v, written deg(v). The maximum degree is ∆(G), the minimum degree is δ(G), and G is regular if ∆(G) = δ(G). It is k-regular if the common degree is k. The neighborhood of v, written N (v), is the set of vertices adjacent to v. In a graph G, contraction of edge e with endvertices u, v is the replacement of u and v with a single vertex whose incident edges are the edges other than e that were incident to u or v. A graph is called a weighted graph if each edge e is assigned a nonnegative real number w(e), called the weight of e. Let G be a weighted graph and H a subgraph of G. The weight of H is defined by w(H) =. X. w(e).. e∈E(H). For graphs G and H, the symmetric difference G ⊕ H is the subgraph of G ∪ H whose edges are the edges of G ∪ H appearing in exactly one of G and H. The symmetric difference of two even subgraphs Z1 and Z2 , is the even subgraph (Z1 ∪ Z2 )\(Z1 ∩ Z2 ). 2.

(16) A matching in a graph G is a set of non-loop edges with no shared endvertices. The vertices incident to the edges of a matching M are saturated by M . A perfect matching in a graph G is a matching that saturates every vertex of G. A factor of a graph G is a spanning subgraph of G. A k-factor is a spanning k-regular subgraph. An odd component of a graph is a component of odd order; the number of odd component of H is o(H). A disconnecting set of edges is a set F ⊆ E(G) such that G − F has more than one component. A graph is k-edge-connected if every disconnecting set has at least k edges. Given S, T ⊆ V (G), we write [S, T ] for the set of edges having one endvertices in S and the other in T . An edge cut is an edge set of the form [S, S], where S is a nonempty proper subset of V (G) and S denotes V (G) − S. In a graph G, subdivision of an edge uv is the operation of replacing uv with a path u, w, v through a new vertex w. A subdivision of H is a graph obtained from a graph H by successive edge subdivisions. An isomorphism from a simple graph G to a simple graph H is a bijection f : V (G) → V (H) such that uv ∈ E(G) if and only if f (u)f (v) ∈ E(H). Two graphs G and G0 are homeomorphic if there is an isomorphism from some subdivision of G to some subdivision of G0 .. 1.2. Preliminaries. First, we introduce the cycle cover and integer flow. Now, we need some definitions and notations. Definition 1.2.1. A cover of a graph G is a collection H of subgraphs of G which covers all edges of G. A cycle cover of a graph G is a collection C of cycles of G which covers all edges of G; C is called a cycle m-cover of G if each edge of G is covered 3.

(17) exactly m times by the members in C. A cover is called an (l, m)-cover if each edge of G is covered either exactly l or exactly m times. Definition 1.2.2. An orientation D of an undirected graph G is an assignment of a direction to each edge e ∈ E(G). Let G be a graph with orientation D. For each vertex v ∈ V (G), E + (v) is the set of non-loop edges with tail v, and E − (v) the set of non-loop edges with head v. A flow in G under orientation D is an integer-valued function φ on E(G) such that X. φ(e) =. X. φ(e) for each v ∈ V (G).. e∈E − (v). e∈E + (v). The support of φ is defined by S(φ) = {e ∈ E(G) : φ(e) 6= 0}. For a positive integer k, if −k < φ(e) < k for every e ∈ E(G), then φ is called a k-flow, and furthermore, if S(φ) = E(G), then φ is called a nowhere-zero k-flow. Definition 1.2.3. For a flow φ in G, we define Eodd (φ) = {e ∈ E(G) : φ(e) is odd} and Eeven (φ) = {e ∈ E(G) : φ(e) is even}. Conjecture 1.2.4. (Tutte’s 3-flow Conjecture) Every 4-edge-connected graph has a nowhere-zero 3-flow. A basic result on integer flows is the following one due to Tutte((6.3) of [22]). Lemma 1.2.5. [22] If G has a flow φ, then for any integer k > 0, G has a k-flow φ0 such that φ0 (e) ≡ φ(e)(mod k) for every e ∈ E(G). 4.

(18) The following is an easy consequence of Lemma 1.2.5. Lemma 1.2.6. Let F ⊆ E(G) and let G0 be obtained from G by contracting the edges in F . If G0 has a k-flow φ0 , then G has a k-flow φ such that S(φ0 ) ⊆ S(φ). A flow in G is always associated with some orientation of G. By changing signs, one can arrange for two flows in G to have the same orientation. If φ1 and φ2 are two flows in G under the same orientation D, then for any integers l and m the linear combination φ = lφ1 + mφ2 is a flow under D. Let φ be a flow in G under a given orientation and e ∈ E(G). If we reverse the direction of e and change φ(e) to −φ(e), we still have a flow in G under the new orientation. Thus, we have Proposition 1.2.7. If G has a flow φ under some orientation D, then for any orientation D0 , G has a flow φ0 under D0 with |φ0 (e)| = |φ(e)| for every edge e. Definition 1.2.8. Let φ be a flow in G. A flow f in G is called a sub-flow of φ if 1. f has the same orientation as φ; 2. |f (e)| ≤ |φ(e)| and f (e)φ(e) ≥ 0 for every e ∈ E(G). Moreover, if f is a k-flow, then we call f a sub-k-flow of φ. By the definition, if f is a sub-flow of φ, then φ − f is also a sub-flow of φ. Lemma 1.2.9. [16] Let φ be a flow in a graph G and r a real number, r ≥ 1. Then c or d φ(e) e for every edge e ∈ E(G). G has a sub-flow f of φ in which f (e) = b φ(e) r r By this result, if a graph G has a k-flow φ, where k ≥ 2, then it has a sub-2-flow f of φ (using r = k − 1). Since φ − f is a sub-(k-1)-flow of φ, we may apply the lemma to φ − f using r = k − 2 (if k ≥ 3). Repeatedly, we decompose the k-flow φ into (k-1) sub-2-flows. This is a result obtained by Little, Tutte, and Younger [17]. We state it as Lemma 1.2.10 below. 5.

(19) Lemma 1.2.10. [17] Every k-flow φ is a sum of (k − 1) sub-2-flows of φ. Lemma 1.2.10 plays a key role in the proof of the next lemma. For a flow φ in a graph G, we set Ei (φ) = {e ∈ E(G) : φ(e) = i} and E±i (φ) = Ei (φ) ∪ E−i (φ). Lemma 1.2.11. [8] Let f be a k-flow in G. Then there is a k-flow φ in G such that S(φ) = S(f ) and |E±1 (φ)| ≥ Proof.. k−1 (|E±1 (f )| + |E±(k−1) (f )|). k. By Lemma 1.2.10, we have that f =. Pk−1 i=1. fi , where fi is a sub-2-flow of. f . Let φi = kfi − f, 1 ≤ i ≤ k − 1. Since fi (e) and f (e) have the same sign for each e ∈ E(G), each φi is a k-flow in G with the same support as f . Let e ∈ E(G); if |f (e)| = 1, then there is exactly one j such that |fj (e)| = 1, and so |φi (e)| = 1 for all i 6= j, 1 ≤ i ≤ k − 1; if |f (e)| = k − 1, then for every i, 1 ≤ i ≤ k − 1, |fi (e)| = 1 and so |φi (e)| = 1. Therefore, k−1 X. |E±1 (φi )| ≥ (k − 2)|E±1 (f )| + (k − 1)|E±(k−1) (f )|.. i=1. That is, |E±1 (f )| +. k−1 X. |E±1 (φi )| ≥ (k − 1)(|E±1 (f )| + |E±(k−1) (f )|).. i=1. Choosing φ ∈ {f, φ1 , ..., φk−1 } with |E±1 (φ)| maximum, k|E±1 (φ)| ≥ (k − 1)(|E±1 (f )| + |E±(k−1) (f )|). Dividing both sides by k yields the required result. Proposition 1.2.12. Let φ be a flow in G and Z = Eodd (φ). Then Z is an even subgraph of G. 6.

(20) Lemma 1.2.13. Let φ be a 4-flow in G. Then G contains two even subgraphs Z1 and Z2 such that Z1 ∪ Z2 = S(φ) and Z1 ∩ Z2 = E±2 (φ). Proof.. We apply Lemma 1.2.9 to φ using r = 2 to obtain a sub-3-flow f , and. let f 0 = φ − f . Then E±1 (f ) and E±1 (f 0 ) are two even subgraphs of the required properties. In second part, we introduce ”Cover by perfect matchings”. Definition 1.2.14. An r-graph G is an r-regular graph such that for each vertex subset X ⊂ V (G) with |X| ≡ 1 (mod 2) and 0 < |X| < |V (G)|, [X, X] ≥ r. Lemma 1.2.15. Let G be an r-regular and r-edge-connected graph then G is an r-graph. The following theorem is a corollary of Edmonds’ Matching Polyhedron Theorem [5]. Theorem 1.2.16. (Edmonds [5]) Let G be an r-graph. Then there is an integer p and a family M of perfect matchings such that each edge of G is contained in exactly p members of M. Theorem 1.2.17. Let G be an r-graph where r ≥ 3 is odd. Then G contains a perfect matching M such that w(M ) ≤ 1r w(G). Proof.. By Theorem 1.2.16, the r-graph G has a family M of perfect matchings. such that each edge of G is contained in exactly p members of M. Thus the number of perfect matchings in M, |M| = rp. Let A be an r-edge-cut of G. For any M ∈ M, |A ∩ (E(G)\M )| is even because E(G)\M is a disjoint union of 2-factors of G, and. 7.

(21) so |A ∩ M | is odd. It follows that |A ∩ M | ≥ 1 for every M ∈ M. Since each edge of A belongs to exactly p members of M, X. |A ∩ M | = rp = |M|.. M ∈M. Consequently, |A ∩ M | = 1 for any r-edge-cut A of G and every M ∈ M. Since each edge of G belongs to exactly p members of M, X. w(M ) = pw(G).. M ∈M. Hence there is some M ∗ ∈ M such that w(M ∗ ) ≤. 1 pw(G) = w(G). |M| r. 8.

(22) Chapter 2 Known Results and Conjectures In this chapter, we introduce several conjectures and known results related to cycle cover and integer flows.. 2.1. Conjectures. The following problem was first considered by Szekeres [20] for bridgeless cubic graphs, and independently, formulated as a conjecture by Seymour [18] for general bridgeless graphs. It is now known as the ”Cycle Double Cover Conjecture”. Conjecture 2.1.1. (Cycle Double Cover Conjecture) Every bridgeless graph has a cycle 2-cover. A stronger form of this conjecture was proposed later by Celmins [4]. Conjecture 2.1.2. (Celmins [4]) Every bridgeless graph has a 2-cover by 5 even subgraphs. If the graph is cubic, then Fulkerson expects that we have better outcome. Conjecture 2.1.3. (Fulkerson [10]) Every bridgeless cubic graph has a 2-cover by 6 perfect matchings. In fact, Conjecture 2.1.3 is equivalent to 9.

(23) Conjecture 2.1.4. Every bridgeless cubic graph has a 4-cover by 6 even subgraphs. Since every bridgeless graph is a contraction of some bridgeless cubic graph (”split” each vertex into a cycle), the following conjecture is equivalent to Conjecture 2.1.4. Conjecture 2.1.5. Every bridgeless graph has a 4-cover by 6 even subgraphs. There are other conjectures on the study of the existence of nowhere-zero flows, we list some of them in what follows. Conjecture 2.1.6. (Goddyn [11]) Every bridgeless graph has a cycle 6-cover. Conjecture 2.1.7. (Tutte) Every bridgeless graph without 3-cuts has a nowhere-zero 3-flow. Conjecture 2.1.8. (Tutte) Every bridgeless graph containing no subgraph contractible to the Petersen graph has a nowhere-zero 4-flow. Conjecture 2.1.9. (Tutte’s 3-flow Conjecture) Every 4-edge-connected graph has a nowhere-zero 3-flow. Conjecture 2.1.10. (Tutte’s 4-flow Conjecture [1966]) Every bridgeless graph containing no subdivision of the Petersen graph has a nowhere-zero 4-flow. Conjecture 2.1.11. (Tutte’s 5-flow Conjecture [1954]) Every bridgeless graph has a nowhere-zero 5-flow. In order to investigate the structure of a cycle cover, Fan started to consider the total size of cycles used in a cover. He obtained a substantial result as we shall see in next section and also posed the following. Conjecture 2.1.12. (Fan [7]) Every 4-edge-connected graph G can be covered by two even subgraphs of total size at most 56 |E(G)|. 10.

(24) The following two conjectures are due to Pyber (Problem 19 in [2]). Conjecture 2.1.13. Every bridgeless graph G has a cycle cover such that every vertex of G is contained in at most ∆(G) cycles of the cover. Conjecture 2.1.14. Every bridgeless graph G has a cycle cover such that every vertex of G is contained in at most 32 ∆(G) + 1 cycles of the cover. Therefore, the Cycle Double Cover Conjecture has the following equivalent formulation. Conjecture 2.1.15. Every bridgeless graph G has a cycle cover such that every vertex v of G is contained in at most d(v) cycles of the cover. All the above conjectures remain open to date.. 2.2. Known results. Bermond, Jackson and Jaeger [3] established the following theorem. Theorem 2.2.1. [3] Every bridgeless graph has a 4-cover by 7 even subgraphs. Godden also made some effort and prove. Theorem 2.2.2. (Goddyn [11]) For each bridgeless graph G, there exists an integer k (depending on |E(G)|) such that G has a cycle (4k + 2)-cover. Theorem 2.2.3. (Goddyn [11]) Every bridgeless graph G has a 6-cover by 10 even subgraphs. Clearly, Theorem 2.2.1 implies that every bridgeless graph has a cycle m-cover for any m ≡ 0(mod 4) and m ≥ 4. For odd m, the following result tells everything.. 11.

(25) Theorem 2.2.4. If m is odd, a graph G has a cycle m-cover if and only if G is an even graph. Proof.. If G is even, then m copies of G give the required cover. Conversely,. suppose that {H1 , H2 , . . . , Hl } is a cycle m-cover of G and m is odd. Then G = H1 ⊕ H2 ⊕ . . . ⊕ Hl and so G is even. Theorem 2.2.3 together with Theorem 2.2.1 yields Theorem 2.2.5. Every bridgeless graph has an m-cover for any even number m ≥ 4. Toward a proof of Conjecture 2.1.11, Jaeger [14] established Theorem 2.2.6. [14] Every bridgeless graph has a nowhere-zero 8-flow. Theorem 2.2.6 was improved by Seymour [19] in the following famous theorem. Theorem 2.2.7. [19] Every bridgeless graph has a nowhere-zero 6-flow. By a result of Jaeger [13] derived from Tutte’s work, every graph with a nowherezero 4-flow can be covered by two even subgraphs, and these two subgraphs together with their symmetric difference, form a 2-cover by three even subgraphs. Thus we have the following result. Theorem 2.2.8. Every graph with a nowhere-zero 4-flow has a 2-cover by 3 even subgraphs. On the size of cover, Fan obtained the following results. Theorem 2.2.9. [7] If G has a nowhere-zero 4-flow, then the minimum total size of two even subgraphs which together cover G is at most |E(G)| + |V (G)| − 1. Theorem 2.2.10. (Fan [6]) Every bridgeless weighted graph G has a (2, 4)-cover by four even subgraphs of total weight at most 12. 20 w(G). 9.

(26) Theorem 2.2.11. (Fan [9]) Every bridgeless graph G has a cycle cover such that each vertex v of G is contained in at most d(v) cycles of the cover if d(v) ≥ 3 and in at most three cycles of the cover if d(v) = 2. Our research is in fact motivated by the following result. We intend to find a cycle cover with minimum total size. Theorem 2.2.12. (Alon and Tarsi [1]) Every bridgeless G has a cycle cover of size at most 35 |E(G)|. An improved version is as follows. Theorem 2.2.13. (Bermond, Jackson and Jeager [3]) Every bridgeless G has a cycle cover of size at most |E(G)| + min{ 23 |E(G)|, 73 (|V (G)| − 1)}. The following result is the best known result so far. Theorem 2.2.14. (Fan [8]) 1. If Tutte’s 3-flow conjecture is true, then every bridgeless graph G has a nowherezero 6-flow φ such that |Eodd (G)| ≥ 23 |E(G)|. 2. If G has a nowhere-zero 6-flow φ such that |Eodd (G)| ≥ 23 |E(G)|, then G has a cycle cover in which the sum of lengths of the cycles in the cycle cover is at most. 44 |E(G)|. 27. 13.

(27) Chapter 3 Main Results and Conclusion 3.1. The Main Results. In [8], Fan proved that if Tutte’s 3-flow Conjecture is true, then every bridgeless graph G has a nowhere-zero 6-flow φ such that |Eodd (G)| ≥ 32 |E(G)| and he utilized this conclusion to show such a graph G has a cycle cover in which the sum of lengths of cycles in the cycle cover is at most. 44 |E(G)|. 27. Clearly, bridgeless graphs are what. he was concerned. In what follows, motivated by this result, we consider the graphs which are (in some sense) more dense. That is, we shall focus on these graphs G with δ(G) = k and G is (k − 1)-edge-connected. We expect that for larger k, the total size of a cycle cover comparing to. 44 |E(G)| 27. will be smaller.. Suppose that F is an even subgraph of a graph G. If G has a 3-flow φ with E(G)\E(F ) ⊆ S(φ), then we let f be a 2-flow in G with S(f ) = F ; it is easy seen that g = 2φ + f is a nowhere-zero 6-flow with Eodd (g) = F . Conversely, if G has a nowhere-zero 6-flow g with Eodd (g) = F , then by a result of Tutte (Lemma 1.2.5), G has a 3-flow φ in which φ(e) = 0 only if |g(e)| = 3, and thus, only if e ∈ E(F ). These results show the following two conjectures are equivalent. Conjecture 3.1.1. Every (k −1)-edge-connected graph G with δ(G) = k has an even subgraph F and 3-flow φ such that E(G)\E(F ) ⊆ S(φ) and |E(F )| ≥ 14. k−1 |E(G)|. k.

(28) Conjecture 3.1.2. Every (k − 1)-edge-connected graph G with δ(G) = k has a nowhere-zero 6-flow g such that |Eodd (g)|≥. k−1 |E(G)|. k. In what follows, we shall first prove the case when k is odd. Theorem 3.1.3. If Tutte’s 3-Flow Conjecture is true, then every graph which is homeomorphic to a (k − 1)-edge-connected graph G with δ(G) = k (k is odd) has an even subgraph F and 3-flow φ such that E(G)\E(F ) ⊆ S(φ) and |E(F )| ≥. k−1 |E(G)|. k. Note that unweighted graph can be regarded as a weighted graph in which each edge e is assigned weight w(e) = 1. On the other hand, for a weighted graph with a weight function w; since the set of rational members are dense, we may assume that w is rational-valued. By multiplying out denominators, we have a weighted graph in which each edge has an integer weight w0 (e). Replacing each edge e by a path of length w0 (e) gives an unweighted graph. By these observation we see that Theorem 3.1.3 is equivalent to Theorem 3.1.4. If Tutte’s 3-Flow Conjecture is true, then every weighted (k − 1)edge-connected graph G with δ(G) = k (k is odd) has an even subgraph F and 3-flow φ such that E(G)\E(F ) ⊆ S(φ) and w(F ) ≥. k−1 w(G). k. The following lemmas are essential to the proof of the main theorem. Lemma 3.1.5. For S ⊆ V (G), let d(S) = |[S, S]|. Let X and Y be nonempty proper vertex subsets of G. Then d(X ∪ Y ) + d(X ∩ Y ) ≤ d(X) + d(Y ). Proof.. Since. d(X) = |[X, X]| = |[X\Y, Y \X]| + |[X\Y, V \(X ∪ Y )]| + |[X ∩ Y, V \(X ∪ Y )]| + |[X ∩ Y, Y \X]|, 15.

(29) d(Y ) = |[Y, Y ]| = |[Y \X, X\Y ]| + |[Y \X, V \(X ∪ Y )]| + |[X ∩ Y, V \(X ∪ Y )]| + |[X ∩ Y, X\Y ]|,. d(X ∩ Y ) = |[X ∩ Y, X\Y ]| + |[X ∩ Y, Y \X]| + |[X ∩ Y, V \(X ∪ Y )]| and d(X ∪ Y ) = |[X\Y, V \(X ∪ Y )]| + |[Y \X, V \(X ∪ Y )]| + |[X ∩ Y, V \(X ∪ Y )]| . Thus, d(X ∪ Y ) + d(X ∩ Y ) ≤ d(X) + d(Y ). Lemma 3.1.6. Let G be an (r−1)-edge-connected graph with δ(G) = r. Then for each v ∈ V (G) with deg(v) > r, there exist two edges uv and vw such that G−uv −vw+uw is (r − 1)-edge-connected. Proof.. Our goal is to find u and w such that G − uv − vw + uw is an (r − 1)-edge-. connected. Fix v ∈ V (G) with deg(v) > r. Among all the edge cuts that contain some edges incident to v, choose S a minimum edge cut so that the resulting disconnected graph G − S has a component S1 with the smallest order, where v ∈ S1 . Since S is a minimum edge cut, then G−S has exactly two components. Let S2 be another component of G − S. Clearly, |S| ≥ r − 1.. Case 1. |S| > r. Select arbitrary two neighbors of v, called u and w. It is easy to see that G − uv − vw + uw is also an (r − 1)-edge-connected graph. 16.

(30) Case 2. |S| = r. Let T = {u ∈ S1 \{v} : u is incident to e ∈ S} and U = {e ∈ S : v is incident to e ∈ S}. Since deg(v) > r and |T | + |U | ≤ |S| = r, there is a vertex x ∈ N (v) ∩ S1 \T . Select y ∈ N (v) ∩ S2 . Claim 1. xv and vy do not belong to any edge cut S 0 with |S 0 | ≤ r.. Suppose not. Let S 0 be an edge cut containing xv and vy, and |S 0 | ≤ r. We only consider |S 0 | = r, since the minimum edge cut of size around v is r. Let K1 be a component of G − S 0 and v ∈ V (K1 ). Moreover, let V (K1 ) = T1 . By Lemma 3.1.5, d(S1 ∩ T1 ) + d(S1 ∪ T1 ) ≤ d(S1 ) + d(T1 ). Since d(S1 ) = |S| = r, d(T1 ) = |S 0 | = r and d(S1 ∪ T1 ) ≥ r, d(S1 ∩ T1 ) ≤ r. Therefore, S1 ∩ T1 ( S1 since u ∈ S1 \T1 . This contradicts to the fact that S1 has the smallest order. ( [T1 , T1 ] is a smaller edge cut. ) Hence S 0 does not exist. Now, by letting u = x and w = y, we conclude the proof.. Case 3. |S| = r − 1. Again, let T = {u ∈ S1 \{v} : u is incident to e ∈ S} and U = {e ∈ S : v is incident to e ∈ S}. Since deg(v) > r and |T |+|U | ≤ |S| = r−1, there is a vertex x ∈ (N (v)∩S1 )\T . Select y ∈ N (v) ∩ S2 . Claim 2. Let S 0 be an edge cut containing xv and vy, then |S 0 | ≥ r.. 17.

(31) Suppose not. Let |S 0 | ≤ r−1, then |S 0 | = r−1. Let K1 be a component of G−S 0 and v ∈ V (K1 ) = T1 . By Lemma 3.1.5, d(S1 ∩T1 )+d(S1 ∪T1 ) ≤ d(S1 )+d(T1 ),and d(S1 ) = |S| = r − 1, d(T1 ) = |S 0 | = r − 1, d(S1 ∪ T1 ) ≥ r − 1, then d(S1 ∩ T1 ) ≤ r − 1. Therefore, S1 ∩ T1 ( S1 since x ∈ S1 \T1 . This contradicts that S1 has the smallest order again. Hence S 0 does not exist.. Following Claim 2., it suffices to consider the case that |S 0 | ≥ r.. Case 3.1. |S 0 | > r. Let x = u and y = w, then the lemma hold. Case 3.2. |S 0 | = r. Among all the r-edge-cuts containing xv, vy, let S 00 be a minimum r-edge-cut so that the resulting disconnected graph G − S 00 has a component K1 with the smallest order, where v ∈ V (K1 ) = T1 . By Lemma 3.1.5, since d(S1 ∩ T1 ) + d(S1 ∪ T1 ) ≤ d(S1 ) + d(T1 ), d(S1 ) = |S| = r − 1, d(T1 ) = |S 0 | = r, and d(S1 ∪ T1 ) ≥ r − 1, d(S1 ∩ T1 ) ≤ r. Since x, y ∈ / S1 ∩ T1 . Therefore, S1 ∩ T1 = T1 which implies T1 ⊆ S1 . Let V. = {u ∈ T1 \{v} : u is incident to e ∈ S 00 } and. W = {e ∈ S 00 : v is incident to e ∈ S 00 }. Since deg(v) > r and |V | + |W | ≤ |S 00 | = r, there is a vertex a ∈ (N (v) ∩ T1 )\V . Claim 3. va and vx do not belong to any edge cut of size not greater than r.. Suppose, to the contrary, the claim is not true. Let S 000 be an edge cut containing va and vx such that |S 000 | ≤ r. Let A1 be a component of G−S 000 and v ∈ A1 .. 18.

(32) Case 1. |S 000 | = r − 1. If |S 000 | = r − 1, then y ∈ A1 . For otherwise, vx and vy are in |S 000 | which implies |S 000 | ≥ r, a contradiction. This contradicts to |S 000 | = r − 1. Since y ∈ A1 , d(A1 ∪ T1 ) ≥ r − 1. By Lemma 3.1.5, since d(A1 ∩ T1 ) + d(A1 ∪ T1 ) ≤ d(A1 ) + d(T1 ), d(T1 ) = |S 00 | = r, d(A1 ) = |S 000 | = r − 1, and d(A1 ∪ T1 ) ≥ r − 1, d(A1 ∩ T1 ) ≤ r. Furthermore x, y ∈ / A1 ∩ T1 therefore A1 ∩ T1 ( T1 since a ∈ T1 \A1 . This contradicts to that T1 has the smallest order. Hence S 000 does not exist.. Case 2. |S 000 | = r. By Lemma 3.1.5, since d(A1 ∩T1 )+d(A1 ∪T1 ) ≤ d(A1 )+d(T1 ), d(T1 ) = |S 00 | = r, d(A1 ) = |S 000 | = r and d(A1 ∪ T1 ) ≥ r, d(A1 ∩ T1 ) ≤ r. By the fact that x, y ∈ / A1 ∩ T1 , this implies that A1 ∩ T1 ( T1 since a ∈ T1 \A1 . This contradicts to that T1 has smallest order. Hence S 000 does not exist either.. Claim 3.1. d(A1 ∪ T1 ) ≥ r.. Since x ∈ / A1 ∪T1 , we only show that vx is not belong to any edge cut of size r−1. Suppose, to the contrary, that the claim is not true. Let E be an edge cut contain vx, and |E| = r − 1. Let K be a component of G − E and v ∈ V (K) = B. Since vx, vy are not belong to any edge cut of size r − 1 at the same time, then y ∈ B. By Lemma 3.1.5, since d(S1 ∩ B) + d(S1 ∪ B) ≤ d(S1 ) + d(B), d(S1 ) = |S| = r−1, d(B) = |E| = r−1, and d(S1 ∪B) ≥ r−1, d(S1 ∩T1 ) ≤ r−1. Therefore, S1 ∩ B ( S1 since x ∈ S1 \B. This contradicts to the fact that S1 has the smallest order. Hence E does not exist.. 19.

(33) In Case 3.2., let a = u and x = w, we yield an (r − 1)-edge-connected graph by G − uv − vw + uw. Combining the above three cases, we complete the proof.. Now, instead of proving Theorem 3.1.3, we prove Theorem 3.1.4. Before going to the proof we need the following fact and a couple of lemmas.. Fact 1. Let G be a (k − 1)-edge-connected graph with δ(G) = k. If G has a vertex y of degree greater than k + 1, then ( by Lemma 3.1.6) there are two edges xy and yz such that deleting xy, yz and joining x and z by a new edge yield an (k − 1)-edge-connected graph. Let G0 be the new graph and assign to the new edge the weight w(xy) + w(yz) so that w(G0 ) = w(G). Then, that an even subgraph of G0 can be extended to an even subgraph of G with the same weight and a 3-flow in G0 can be extended to a 3-flow in G with the same support.. Lemma 3.1.7. If Tutte’s 3-Flow Conjecture is true, then every weighted (k − 1)edge-connected graph G (k is odd) has an even subgraph F and a 3-flow φ such that E(G)\E(F ) ⊆ S(φ) and w(F ) ≥ Proof.. k−1 w(G). k. We use induction on |E(G)|. Basic step: Consider G is (k − 1)-regular. graph and (k − 1)-edge-connected. Since k is odd, G is an even graph then G satisfies the properties. Induction step: We assume that the claim holds for |E(G)| ≤ r. Consider |E(G)| = r + 1. Let G have a (k − 1)-edge-cut {e1 , e2 , . . . , ek−1 }, G0 be the graph obtained by contracting e1 and eˆ1 be the contracted vertex. Reassign to ei a. 20.

(34) new weigth w(e1 ) + w(ei ) so that w(G0 ) = w(G), where ei is the edge close to e1 in even subgraph of G0 . Consider an even subgraph F 0 of G0 and a 3-flow φ0 in G0 . If eˆ1 ∈ / F 0 , then F 0 gives an even subgraph of G with the same weight set of edges, therefore wG (F 0 ) = wG0 (F 0 ) ≥. k−1 w(G0 ) k. =. k−1 w(G); k. if eˆ1 ∈ F 0 , then F 0 ∪ {e1 } is an. even subgraph of G with the same weight as F 0 , therefore, wG (F 0 ∪{e1 }) = wG0 (F 0 ) ≥ k−1 w(G0 ) k. =. k−1 w(G). k. Moreover, in either case the 3-flow φ0 can be extended to a. 3-flow φ in G in which φ(e) = φ0 (e) for all e ∈ E(G)\{e1 } and φ(e1 ) = φ0 (ei ) or −φ0 (ei ), according to the orientation of ei . Then G satisfies the properties. If G has a vertex y of degree greater than k + 1, then by Fact 1. G satisfies the properties of Theorem 3.1.4. If G have a (k − 1)-edge-cut, then by Lemma 3.1.7 G satisfies the properties of Theorem 3.1.4. Therefore, if G is a counterexample to the statement of Theorem 3.1.4 with a minimum number of edges, then G satisfies the following lemma. Lemma 3.1.8. If G is a counterexample to the statement of Theorem 3.1.4 with a minimum number of edges, then G is simple, k-regular and k-edge-connected. Proof of Theorem 3.1.4.. Suppose, to the contrary, that the theorem is not true.. Let G be a minimum counterexample. Then, by Lemma 3.1.8, G is a simple, k-regular and k-edge-connected. By Theorem 1.2.17, G contains an even subgraph F with w(F ) ≥. k−1 w(G) k. such that the graph G0 obtained by contracting each component of. F is 4-edge-connected. If Tutte’s 3-flow conjecture is true, then G0 has a nowhere-zero 3-flow φ with E(G0 ) ⊆ S(φ). But E(G0 ) = E(G)\E(F ), and so F and φ have the required properties. This contraction proves Theorem 3.1.4. Now we return to unweighted graphs. Before proving our main result, we need the following lemma. 21.

(35) Lemma 3.1.9. Let g be a 6-flow in G. Then there is a 6-flow φ in G such that S(φ) = S(g) and 5 |E±1 (φ)| ≥ |Eodd (g)|. 9 Proof.. Since Eodd (g) is an even subgraph of G, we may define a 2-flow f0 in G. with S(f0 ) = Eodd (g). Applying Lemma 1.2.5 to the flows g + 2f0 and g − 2f0 yields two 6-flows in G, say f1 and f2 , respectively, in which for every e ∈ E(G) f1 (e) ≡ (g(e) + 2f0 (e)) (mod 6) and f2 (e) ≡ (g(e) − 2f0 (e)) (mod 6). It is not difficult to see that E±3 (f1 ) ∪ E±3 (f2 ) = E±1 (g) ∪ E±5 (g). Adding E±3 (g) to both sides yields E±3 (g) ∪ E±3 (f1 ) ∪ E±3 (f2 ) = Eodd (g). Let f ∈ {g, f1 , f2 } with |E±3 (f )| smallest. Then 1 |E±3 (f )| ≤ |Eodd (g)|. 3 Since f is a 6-flow in G, by Lemma 1.2.11 with k = 6, there is a 6-flow φ in G such that S(φ) = S(f )(= S(g)) and 5 5 (|E±1 (f )| + |E±5 (f )|) = (|Eodd (f )| − |E±3 (f )|) 6 6 1 5 (|Eodd (f )| − |Eodd (g)|). ≥ 6 3. |E±1 (φ)| ≥. Since Eodd (f ) = Eodd (g), 5 5 2 |E±1 (φ)| ≥ ( |Eodd (g)|) = |Eodd (g)|. 6 3 9 This completes the proof. 22.

(36) Theorem 3.1.10. If a (k − 1)-edge-connected graph G with δ(G) = k (k is odd) has a nowhere-zero 6-flow φ such that |Eodd (φ)|≥ which the size of the cycle cover is at most Proof.. k−1 |E(G)|, k. then G has a cycle cover in. 13k+5 |E(G)|. 9k. Let g be a nowhere-zero 6-flow in G with |Eodd (g)| ≥. k−1 |E(G)|. k. It follows. from Lemma 3.1.9 that G has a nowhere-zero 6-flow φ such that 5(k − 1) 5 |E(G)|. |E±1 (φ)| ≥ |Eodd (g)| ≥ 9 9k Applying Lemma 1.2.9 to φ with r = 2, we have a sub-4-flow of φ, say φ1 , such that φ1 (e) = b φ(e) c or d φ(e) e, for every e ∈ E(G). Set φ2 = φ − φ1 . Then φ2 is also a 2 2 sub-4-flow of φ. Let A = {e ∈ E(G) : φ1 (e) = 0} and B = {e ∈ E(G) : φ2 (e) = 0}. Then A ∩ B = ∅ and A ∪ B = E±1 (φ). Moreover, |φ1 (e)| = 1 if e ∈ B and |φ2 (e)| = 1 if e ∈ A. Applying Lemma 1.2.13 to φ1 , we have two even subgraphs X1 and X2 , such that X1 ∪ X2 = S(φ1 ) and X1 ∩ X2 = E±2 (φ1 ). Set X3 = X1 ⊕ X2 . Then B ⊆ X3 and {X1 , X2 , X3 } covers each edge of S(φ1 ) = E(G) − A exactly twice. Similarly, φ2 yields three even subgraphs, say Y1 , Y2 , and Y3 , such that A ⊆ Y3 and {Y1 , Y2 , Y3 } covers each edge of S(φ2 ) = E(G) − B exactly twice. Let C1 = {X1 , X2 , Y3 } and C2 = {Y1 , Y2 , X3 }. Then both C1 and C2 are cycle covers of G. Since 3 X i=1. |Xi | +. 3 X. |Yi | = 2|S(φ1 )| + 2|S(φ2 )| = 4|E(G)| − 2|A ∪ B|. i=1. = 4|E(G)| − 2|E±1 (φ)|, either C1 or C2 is of size at most 2|E(G)| − |E±1 (φ)| ≤ 2|E(G)| −. 13k + 5 5(k − 1) |E(G)| = |E(G)|. 9k 9k. Next, we consider the case when k is even. 23.

(37) Theorem 3.1.11. If Tutte’s 3-Flow Conjecture is true, then every graph which is homeomorphic to a (k − 1)-edge-connected graph G with δ(G) = k (k is even) has an even subgraph F and 3-flow φ such that E(G)\E(F ) ⊆ S(φ) and |E(F )| ≥. k−2 |E(G)|. k−1. Similarly, we see that Theorem 3.1.11 is equivalent to Theorem 3.1.12. If Tutte’s 3-Flow Conjecture is true, then every weighted (k − 1)edge-connected graph G with δ(G) = k (k is even) has an even subgraph F and 3-flow φ such that E(G)\E(F ) ⊆ S(φ) and w(F ) ≥. k−2 w(G). k−1. Before the proof of Theorem 3.1.12 we need the following Lemma. Lemma 3.1.13. If Tutte’s 3-Flow Conjecture is true, then every weighted (k − 1)edge-connected graph G with (k − 1)-edge-cut (k is even) has an even subgraph F and 3-flow φ such that E(G)\E(F ) ⊆ S(φ) and w(F ) ≥ Proof.. k−2 w(G). k−1. By induction on |E(G)|. Basic step: Consider G is (k − 1)-regular graph. and (k − 1)-edge-connected. By Theorem 1.2.17, G contains an even subgraph F with w(F ) ≥. k−2 w(G) k−1. such that the graph G0 obtained by contracting each component of. F is 4-edge-connected. If Tutte’s 3-flow conjecture is true, then G0 has a nowhere-zero 3-flow φ with E(G0 ) ⊆ S(φ). E(G0 ) = E(G)\E(F ), and so F and φ have the required properties. Induction step: We assume that the claim holds for |E(G)| ≤ r. Consider |E(G)| = r + 1. Let G have a (k − 1)-edge-cut {e1 , e2 , . . . , ek−1 }, G0 be the graph obtained by contracting e1 and eˆ1 be the contracted vertex. Reassign to ei a new weigth w(e1 ) + w(ei ) so that w(G0 ) = w(G), where ei is the edge close to e1 in even subgraph of G0 . Consider an even subgraph F 0 of G0 and a 3-flow φ0 in G0 . If eˆ1 ∈ / F 0, then F 0 gives an even subgraph of G with the same weight set of edges, therefore wG (F 0 ) = wG0 (F 0 ) ≥. k−2 w(G0 ) k−1. =. k−2 w(G); k−1. if eˆ1 ∈ F 0 , then F 0 ∪ {e1 } is an even. subgraph of G with the same weight as F 0 , therefore, wG (F 0 ∪ {e1 }) = wG0 (F 0 ) ≥ 24.

(38) k−2 w(G0 ) k−1. =. k−2 w(G). k−1. Moreover, in either case the 3-flow φ0 can be extended to a. 3-flow φ in G in which φ(e) = φ0 (e) for all e ∈ E(G)\{e1 } and φ(e1 ) = φ0 (ei ) or −φ0 (ei ), according to the orientation of ei . Therefore, G satisfies the properties. Proof of Theorem 3.1.12.. Suppose, to the contrary, that the theorem is not. true. Let G be a minimum counterexample. Then, by Lemma 3.1.8, G is a simple, k-regular and k-edge-connected. Since k is even, G is an even graph then G satisfies the properties. This contraction proves Theorem 3.1.12.. Theorem 3.1.14. If a (k − 1)-edge-connected graph G with δ(G) = k (k is even) has a nowhere-zero 6-flow φ such that |Eodd (φ)|≥ which the size of the cycle cover is at most Proof.. k−2 |E(G)|, k−1. then G has a cycle cover in. 13k−8 |E(G)|. 9(k−1). Let g be a nowhere-zero 6-flow in G with |Eodd (g)| ≥. k−1 |E(G)|. k. It follows. from Lemma 3.1.9 that G has a nowhere-zero 6-flow φ such that 5 5(k − 2) |E±1 (φ)| ≥ |Eodd (g)| ≥ |E(G)|. 9 9(k − 1) Applying Lemma 1.2.9 to φ with r = 2, we have a sub-4-flow of φ, say φ1 , such that c or d φ(e) e, for every e ∈ E(G). Set φ2 = φ − φ1 . Then φ2 is also a φ1 (e) = b φ(e) 2 2 sub-4-flow of φ. Let A = {e ∈ E(G) : φ1 (e) = 0} and B = {e ∈ E(G) : φ2 (e) = 0}. Then A ∩ B = ∅ and A ∪ B = E±1 (φ). Moreover, |φ1 (e)| = 1 if e ∈ B and |φ2 (e)| = 1 if e ∈ A. Applying Lemma 1.2.13 to φ1 , we have two even subgraphs X1 and X2 , such that X1 ∪ X2 = S(φ1 ) and X1 ∩ X2 = E±2 (φ1 ). Set X3 = X1 ⊕ X2 . Then B ⊆ X3 and {X1 , X2 , X3 } covers each edge of S(φ1 ) = E(G) − A exactly twice. Similarly, φ2 yields three even subgraphs, say Y1 , Y2 , and Y3 , such that A ⊆ Y3 and {Y1 , Y2 , Y3 } covers each edge of S(φ2 ) = E(G) − B exactly twice. Let C1 = {X1 , X2 , Y3 } and. 25.

(39) C2 = {Y1 , Y2 , X3 }. Then both C1 and C2 are cycle covers of G. Since 3 X i=1. |Xi | +. 3 X. |Yi | = 2|S(φ1 )| + 2|S(φ2 )| = 4|E(G)| − 2|A ∪ B|. i=1. = 4|E(G)| − 2|E±1 (φ)|, either C1 or C2 is of size at most 2|E(G)| − |E±1 (φ)| ≤ 2|E(G)| −. 3.2. 13k − 8 5(k − 2) |E(G)| = |E(G)|. 9(k − 1) 9(k − 1). Conclusion. As can be seen in the main results in Chapter 3, if k = 3, 4, then our result is exactly the same as obtained by Fan 2.2.14 [8]. Therefore, with this technique, we are not able to get a better upper bound for the total lengths of cycles used in covering the graph G (bridgeless). We do believe that this bound. 44 |E(G)| 27. can be. smaller, but not able to prove it at this moment. Since our result is obtained also under the assumption that Tutte’s 3-Flow Conjecture is true, to prove this long-stand conjecture deserves to be considered as the first major goal of our future study.. 26.

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