Hamilton weight and Petersen minor

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1 Hamilton weight and Petersen minor Hong-Jian Lai and Cun-Quan Zhang Department of Mathematics West Virginia University Morgantown, WV , USA Abstract A (1, 2)-eulerian weight w of a cubic graph is hamiltonian if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen-minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K 4 by a series of Y -operations. As a byproduct of the proof of the main theorem, we also prove that if G is a permutation graph and w is a (1, 2)-eulerian weight of G such that (G, w) is a critical contra pair, then the Petersen minor appears almost everywhere in the graph G. 1 Introduction We assume familiarity of graph theory. A circuit is a 2-regular connected graph. By G = H we mean that the graphs G and H are isomorphic. A graph H is called a minor of a graph G if H is isomorphic to a contraction of a subgraph of G. A Petersen-minor is a minor isomorphic to the Petersen graph. A path P = v 1 v p (p 3) of a graph G is called a subdivided edge of G if d G (v i ) = 2 for each v i with 2 i p 1. The underlying graph of a graph G, denoted by G, is the graph obtained from G by replacing each maximal subdivided edge with a single edge. Let H 1 and H 2 be two subgraphs of a graph G. The symmetric difference of H 1 and H 2, denoted by H 1 H 2, is the subgraph of G induced by edges of [E(H 1 ) E(H 2 )] \ [E(H 1 ) E(H 2 )]. A map w : E(G) {1, 2} is called a (1, 2)-weight of G. A graph G associated with a weight w is usually denoted by an ordered pair (G, w). A (1, 2)-weight of a graph G is This research was partially supported by the National Science Foundation under Grant DMS and by the National Security Agency under Grant MDA

2 eulerian if the total weight of each edge-cut of G is even. Let G be a bridgeless graph and with a (1,2)-weight w. A family F of circuits of G is called a faithful circuit cover of (G, w) if each edge e of G is contained in precisely w(e) members of F. Let w be a (1, 2)-eulerian weight of a cubic graph G. A faithful circuit cover F of (G, w) is hamiltonian if F is a set of two Hamilton circuits. A (1, 2)-eulerian weight w of G is hamiltonian if every faithful circuit cover of (G, w) is hamiltonian. If (G, w) has a faithful circuit cover which is not a Hamilton cover, then (G, w) is nonhamiltonian coverable. A cubic graph G is uniquely edge-3-colorable if there is only one way to partition E(G) into 3 perfect matchings of G. In other words, G has only one 1 factorization. The Y -operation is an operation of a cubic graph that contracts a triangle to a vertex. The Y -operation is an operation of a cubic graph that expands a vertex to a triangle (see Figure 1). It is well known that Y -operations on a cubic graph preserves the property of being uniquely edge-3-colorable. Figure 1: Y -operation and Y -operation Let S be the set of all cubic graphs which can be obtained from K2 3 by a series of Y or Y operations (where K2 3 is the graph with 2 vertices and 3 parallel edges). The study of Hamilton weight and Hamilton cover is motivated by and intimately related to the following well-known conjectures. Conjecture A (Circuit double cover conjecture, Szekeres [16] and Seymour [15], or see [13]) Every bridgeless graph has a family of circuits that covers each edge precisely twice. Conjecture B (Unique edge-3-coloring conjecture, Fiorini and Wilson [5] [6]) Let G be a cubic planar graph. Then G is uniquely edge-3-colorable if and only if G S. Prior studies on the problem of unique edge-3-colorings of cubic graphs can be seen, for example, in [5], [6], [10], [11], [17], [18], [19], [20], among others. It is well-known that a smallest counterexample G to Conjecture A can be assumed to be cubic and cyclically 4-edge-connected (see [15] and [13]). Let e E(G) and choose a double cover F of G \ {e} such that F is as large as possible. It is evident that the underlying graph of C C, for each pair of adjacent members of F, admits a Hamilton weight. Thus, 2

3 it has been expected that the structure of C C would provide some powerful tools to the final solution of Conjecture A and some other related circuit covering problems. It was proved in [21] that a 3-connected cubic graph containing no Petersen minor and admitting a Hamilton weight must be uniquely edge-3-colorable. An analog of Conjecture B was also proposed in that paper (Conjecture 4.5 of [21]) that every 3-connected cubic graph G admitting a Hamilton weight is in S. This conjecture is partially proved in this paper for Petersen-minor free graphs (Theorem 1.1). Theorem 1.1 Let G be a 3-connected cubic graph containing no Petersen-minor. Then G admits a Hamilton weight if and only if G S. It is necessary for G to be 2-connected to admit a Hamilton weight. Let (G 1, w 1 ) and (G 2, w 2 ) be two cubic graphs with (1, 2)-weights w 1 and w 2, respectively. For each i {1, 2}, let e i = x i y i E(G i ) with w i (e i ) = 2. Define a new graph G 1 (e1,e 2 ) G 2 from the disjoint union of G 1 e 1 and G 2 e 2 by adding new edges e x = x 1 x 2 and e y = y 1 y 2. Then the following characterizes graphs admitting a Hamilton weight among 2-connected cubic graphs without a Petersen minor. Corollary 1.2 Let G be a 2-connected cubic graph containing no Petersen-minors. Then G admits a Hamilton weight if and only if either G S or for some integer t 1, there exist graphs G 1, G 2,, G t+1 S such that G = ( ((G 1 (e (1) 1,e(1) 2 ) G 2) (e (2) 1,e(2) 2 ) G 3) (e (3) 1,e(3) 2 ) ) (e (t) 1,e(t) 2 ) G t+1. The proof of the main results will be given in the last section. In Section 2, we present some preliminaries on Y -operations and on faithful circuit coverings. An associate result will be proved in Section 3, which is needed in Section 4. 2 Preliminaries Let i 0 be an integer. For a (1, 2)-weight w of a graph G, denote E w=i (G) = {e E(G) : w(e) = i}. When there is no confusion arises, we shall write E w=i for E w=i (G). Some straightforward properties about S are listed below, the proofs of which are omitted or sketched. Lemma 2.1 The Y -operation of a cubic graph preserves the property of admitting a Hamilton weight, and also preserves the number of 1 factorizations of the graph. 3

4 Lemma 2.2 Each G S has a unique 1 factorization. Lemma 2.3 For a graph G S and w be a (1, 2)-eulerian weight of G with E w=2 = M. Then w is a Hamilton weight of G if and only if M is a 1-factor with M T = 1 for each 3-edge-cut T of G. Proof. Apply Y -operations and argue by induction on V (G). Lemma 2.4 ([9]) Every graph G S can be obtained from K 3 2 operations. by a series of only Y Definition 2.5 Let n 0 and k 0 be integers, and let φ : {0, 1, 2,, n} {x, y, z} be a function. Define Γ 1 = Γ 1 (n, φ) to be graph obtained from the vertex disjoint paths P x = x 0 x 1 x n x 0, P y = y 0 y 1 y n y 0 and P z = z 0 z 1 z 2 z n z 0 such that the vertex set and the edge set of Γ 1 are V and E, respectively given below: V = V (P x ) V (P y ) V (P z ) and E = E(P x ) E(P y ) E(P z ) {x 0 y 0, y 0 z 0, z 0 x 0, x 0y 0, y 0z 0, z 0x 0} {x i y i 1 i n and φ(i) = z} {y i z i 1 i n and φ(i) = x} {z i x i 1 i n and φ(i) = y}. Let Γ(n, φ) be the underlying cubic graph of Γ 1. Note that the range of φ has only one element if and only if there is an edge joining the two triangles x 0 y 0 z 0 x 0 and x 0 y 0 z 0 x 0 in Γ(n, φ). For m 3, let Λ(m) be the graph obtained from a circuit C 2m = v 1 v 2 v 2m v 1 by adding the new edges {v 1 v m+1 } {v i v 2m+2 i 2 i m 1}. Lemma 2.6 Let G S \ {K2 3, K 4}. Each of the following holds. (1) G has at least two triangles. (2) Every pair of distinct triangles of G are disjoint. (3) If, in addition, G has exactly two triangles, then G = Γ(n, φ), for some n and φ. (4) If G = Γ(n, φ) for some n 0 and φ, and if the range of φ has only one element, that is, the only two triangles are joined by an edge, then G = Λ(n + 3). Proof. Parts (1) and (2) follow from Lemma 2.4, and Part (4) follows from Part (3). It suffices to apply induction to prove Part (3). Suppose that T 1 and T 2 are the only two triangles of G and let G 1 be the graph obtained from G by contracting T 1. If G 1 = K4, then G = Γ(0, φ), the only 6-vertex graph in S. 4

5 Hence we may assume that G 1 K 4. By induction, G 1 = Γ(n 1, φ1 ) for some φ 1. We shall use the notation for this Γ in the definition above. Let v T denote the contraction image of the triangle T 1 in V (Γ). Then we may assume that v T {x 0, y 0, z 0 }. Without loss of generality, we assume that v T = y 0. Thus G = Γ(n, φ) where φ(i) = φ 1 (i) if i < n, and φ(n) = y, and so Part (3) is proved by induction. Lemma 2.7 Let G 1 and G 2 be connected cubic graphs, and for i {1, 2}, let v i V (G i ) be a vertex with neighbors x i, y i, z i in G i. Obtain a new graph G = G 1 (v1,v 2 ) G 2 from the disjoint union of G 1 \ v 1 and G 2 \ v 2 by adding the new edges x 1 x 2, y 1 y 2, z 1 z 2. Then G S if and only if both G 1 and G 2 are in S. Proof. We argue by induction on V (G). The lemma holds trivially if both G 1, G 2 {K2 3, K 4} for sufficiency and if V (G) 6 for necessity. Assume that G 1 S \ {K2 3, K 4}. Then G 1 \ v 1 has a triangle T. Apply a Y - operation to G 1 to get G 1 by contracting T. By induction, G = G 1 (v 1,v 2 ) G 2 S. But then G can be obtained from G by applying a Y -operation to restore T, and so G S. This proves the sufficiency. If G S, then G has a triangle T. Since E(T ) {x 1 x 2, y 1 y 2, z 1 z 2 } =, we may assume that T is a subgraph of G 1. Apply Y -operation to G by contracting T, resulting a graph G. Then the same Y -operation is also applied to G 1, resulting a graph G 1 with G = G 1 G 2. Since G S, G S also. By induction, both G 1 and G 2 are in S. Apply a Y -operation to G 1 to restore T, G 1 S, and so this proves the necessity. The following lemma is essentially proved by the same argument of Seymour in the proof for (3.5) of [15]. Lemma 2.8 ([15]) Let w be a (1, 2)-eulerian weight of a 2-connected cubic graph G. If E w=1 (G) is a Hamilton circuit of G, then (G, w) has a faithful circuit cover. Theorem 2.9 ([1] or [2]) Let w be a (1, 2)-eulerian weight of a 2-connected cubic graph G. If G contains no Petersen-minor, then (G, w) has a faithful circuit cover. Lemma 2.10 Let G be a cubic graph admitting a Hamilton weight w. Let {H 1, H 2 } be a faithful Hamilton cover of (G, w). Then each of the following holds. (1) For each circuit C of E w=1, {H 1 C, H 2 C} is also a faithful Hamilton cover of (G, w). (2) If, in addition, G has a 2-edge-cut T, then T E w=2 and E w=1 has more than one component. 5

6 Definition 2.11 Let x and y be two vertices of a graph G. A family C of circuits of G is called a circuit chain joining x and y if C = {C 1,, C p } such that (i) x V (C 1 ) and y V (C p ) (ii) V (C i ) V (C j ) if and only if i = j ± 1. Lemma 2.12 ([21], or see [22]) Let w be a Hamilton weight of a cubic graph G. If the graph G is 3-connected and contains no subdivision of the Petersen graph. Then, for each e 0 = xy E(G) with w(e 0 ) = 2, every faithful circuit cover of (G \ {e 0 }, w) is a circuit chain joining x and y. Proof. For each weight two edge e 0 = xy, by Theorem 2.9, (G \ {e 0 }, w) has a faithful circuit cover F. Let P = {C 1,, C r } be a circuit chain (each C i F) joining the vertices x and y in G. Let H be the graph induced by edges covered by circuits of P and the edge e 0, and let w be a (1, 2)-eulerian weight on E(H) such that w (e) is the number of circuits of P containing the edge e for each e e 0, and w (e 0 ) = 2. By Theorem 2.9, H has a faithful circuit cover F. Then F [F \ P] is a faithful circuit cover of G. If F P, then the faithful circuit cover F [F \ P] of (G, w) is not hamiltonian, contrary to the assumption that w is a Hamilton weight. Therefore F = P. Lemma 2.13 Let (G, w) be a cubic graph with a (1,2)-weight, let e = xy E(G) be an edge with w(e) = 2 and let G = G \ {e}. Suppose that (G \ {e}, w) has a faithful circuit cover F = {C 1, C 2,, C t } which is a circuit chain joining x and y. Then each of the following holds. (i) Each component of E w=1 (G ) is an even circuit. (ii) If, in addition, E w=1 (G) is a Hamilton circuit C of G such that C {e} is a union of 2 even (odd, respectively) circuits whose intersection is {e}, then t is odd (even, respectively). Proof. Let X 0 = {f E(C j ) : j is even } and let X 1 = {f E(C j ) : j is odd }. Then color the edges in X 0 \ X 1 red, the edges in X 1 \ X 0 blue, and the edges in X 0 X 1 yellow. This defines a proper 3-edge-coloring of the underlying graph G = G \ {e 0 }. Since each component of E w=1 (G ) is alternatively colored red and blue, it must be of even length. This proves (i). Now assume that E w=1 (G) is a Hamilton circuit C. Let the subdivided edges of G \ {e} that contain the endvertices of e be e and e. Suppose that C {e} is a union of two circuits of even (odd, respectively) lengths whose intersection is {e}. Since the edges in E(C) = E w=1 (G) are alternatively red and blue colored, e and e must receive same (different, respectively) colors. Note that one of {e, e } is in C 1 \ i 2 E(C i ) and the other is in C t \ i t E(C i ), and so t must be odd (even, respectively). 6

7 Lemma 2.14 ([21], or see [22]) If a 3-connected cubic graph G admits a Hamilton weight w and contains no subdivision of the Petersen graph, then the subgraph of G induced by edges of E w=1 is a Hamilton circuit. Proof. Let w be a Hamilton weight of G and {H 1, H 2 } be a Hamilton cover of (G, w). Note that {H 1, H 2 } induces an edge-3-coloring {H 1 \ H 2, H 2 \ H 1, H 1 H 2 } of G, and so each component of the 2-factor E w=1 = H 1 H 2 is a circuit of even length. We want to show that E w=1 has onle one component. Since any two components of E w=1 are joined in G by an edge in E w=2, it suffices to show that if e 0 = xy E w=2, then x and y must be in the same component of E w=1. Since G contains no subdivisions of the Petersen graph, by Theorem 2.9, (G \ {e 0 }, w) has a faithful circuit cover F = {C 1,, C r }, which, by Lemma 2.12, is also a circuit chain joining x and y. By Lemma 2.13(i), each component of E w=1 not containing x nor y is an even circuit. If x and y are contained in different circuits of E w=1, then E w=1 would have at least two circuits of odd lengths, contrary to the fact that every circuit in E w=2 has even length. Hence x and y are in the same component of E w=1. Therefore E w=1 has only one component, and so E w=1 is a Hamilton circuit of G. Lemma 2.15 (Lemma 2.1 of [21]) Let G be an edge-3-colorable cubic graph, w be a (1, 2)- eulerian weight of G, and M be a 1 factorization of G. If E w=2 / M, then (G, w) is nonhamiltonian coverable. Lemma 2.16 If a 3-connected cubic graph G admits a Hamilton weight w and the subgraph of G induced by edges of E w=1 is a Hamilton circuit, then G is not bipartite. Proof. We argue by contradiction and assume that G is bipartite. Let C = v 1 v 2n v 1 be the Hamilton circuit induced by E w=1. Then, for each e = v i v j E w=2, i and j must have different parity since G has no circuit of odd lengths. Let v 1 v 2k E w=2. Then M = {v i v i+1 : i is odd and 2k+1 i 2n 1} {v 1 v 2k } {v j v j+1 : i is even and 2 j 2k 2} is a perfect matching of G. Since G is bipartite, E(G)\M is a 2-factor with each component as an even length circuit. So, G has a 1-factorization M = {M, M, M } containing M as a member. Note that E w=2 M and E w=2 \ M. Thus E w=2 M, contrary to Lemma For a Hamilton circuit C = v 1 v 2n v 1 of a cubic graph G, a chord v i v j of C is of pace h where h is the minimal of j i and 2n j i. Thus, by Lemma 2.16, we have the following lemma. 7

8 Lemma 2.17 If a 3-connected cubic graph G admits a Hamilton weight w and the subgraph of G induced by edges of E w=1 is a Hamilton circuit C = v 1 v 2n v 1, then some chord e E w=2 of C must be of even pace. Lemma 2.18 If a 3-connected cubic graph G admits a Hamilton weight w and the subgraph of G induced by edges of E w=1 is a Hamilton circuit, then, for a chord e E w=2 of C with even pace, (G \ {e}, w) has a faithful circuit cover and, furthermore, each faithful circuit cover of (G \ {e}, w) must be a circuit chain of even length. Proof. Let C be the Hamilton circuit of G induced by E w=1. First we prove that (G\{e}, w) has a faithful circuit cover. Since the Hamilton circuit C is still a Hamilton circuit in the underlying cubic graph G of G\{e}, it is obvious that G is edge-3-colorable. By Lemma 2.8, (G, w) has a faithful circuit cover. By Lemma 2.12, each faithful circuit cover of (G\{e}, w) must be a circuit chain joining the endvertices of e. By Lemma 2.13 and by the assumption that e has even pace, such a circuit chain must have even length. 3 Nonhamiltonian coverable graphs Let F be a family of subgraphs of a graph G. The weight w F : E(G) {0, 1, 2, } with w F (e) equal to the number of members of F containing e, for each edge e E(G), is called the weight of G induced by F. The main result of this section, Theorem 3.5, plays a key role in Section 4, and is believed to have further applications in the studies of some related problems. We start with some preparations. Definition 3.1 Let G be a cubic graph, w be a (1, 2)-eulerian weight of G, and e 1 and e 2 E w=1. The graph H(G; e 1, e 2 ) is obtained from G by inserting a new vertex, say x i, into each e i {e 1, e 2 } and adding a new edge e joining x 1 and x 2. The weight of the new graph H(G; e 1, e 2 ) (also denoted by w, for convenience) coincides with the original weight with w(e ) = 2. Definition 3.2 Let G be a cubic graph, w be a (1, 2)-eulerian weight of G, and e 1 and e 2 E w=1. The edges e 1 and e 2 are eventually adjacent if e 1 and e 2 become adjacent to each other after a series of Y -operations of G. Lemma 3.3 Let e 1, e 2 be two distinct edges of a graph G S \ {K 3 2, K 4}. Then the edges e 1 and e 2 are not eventually adjacent to each other if one of the followings happens. 8

9 (1) e 1 and e 2 are contained in the same member of the unique 1-factorization of G; (2) e 1 and e 2 are contained in two distinct triangles of G; (3) e 1 is contained in a triangle of G and e 1 is not adjacent to e 1. Proof. (1) is obvious. (2) is a corollary of (2) of Lemma 2.6. The following is the proof for (3). We prove by induction on V (G). It is obvious for V (G) = 6. Let e 1 = u 1 u 2 be contained in a triangle C 1 = u 1 u 2 u 3 u 1 and, by (1), we assume that e i M i for i = 1, 2 where {M 1, M 2, M 3 } is the 1-factorization of G. Let v i / V (C 1 ) (i = 1, 2, 3) be the vertex of G adjacent to u i. Since e 2 is not adjacent to e 1 = u 1 u 2 and e 2 / M 1, e 2 / E(C 1 ) {u 1 v 1, u 2 v 2, u 3 v 3 }. By (1) of Lemma 2.6, let C 2 be a triangle of G distinct from C 1. By (2), we assume that e 2 / E(C 2 ). Let G be the graph obtained from G by contracting C 2. By (2) of Lemma 2.6, no edge of E(C 1 ) {u 1 v 1, u 2 v 2, u 3 v 3, e 2 } is contained in C 2. Thus, e 2 remains disjoint with e 1 and C 1 remains as a triangle in G. The lemma is proved by the induction. Lemma 3.4 Let G S and e 1, e 2 E(G). Then H(G; e 1, e 2 ) S if and only if e 1 and e 2 are eventually adjacent. Proof. follows from Lemma 2.4 and from the definition of Y. : Let G be a smallest counterexample to the lemma. It is obvious that G K 3 2. Thus, H(G; e 1, e 2 ) S \ {K 3 2, K 4}. Let x 1, x 2 be the two new vertices introduced by subdividing e 1 and e 2 when defining H(G; e 1, e 2 ). If H(G; e 1, e 2 ) has a triangle T containing neither x 1 nor x 2, T is also a triangle of G. Applying a Y -operation by contracting T to a vertex in G and H(G; e 1, e 2 ), we obtain new graphs G and H respectively, where H = H(G ; e 1, e 2 ). Since H S and since G is a smallest counterexample to the lemma, the edges e 1, e 2 are eventually adjacent in G, so are in G. Thus, we may assume that every triangle of H(G; e 1, e 2 ) contains either x 1 or x 2. Hence H(G; e 1, e 2 ) has precisely two disjoint triangles, and so by Lemma 2.6, each of which contains exactly one of x 1 and x 2. It follows that these two triangles are 2-circuits in G, contrary to the fact that no graph in S \ {K2 3 } has a 2-circuit. Theorem 3.5 Let G S, w : E(G) {1, 2} be a Hamilton weight of G, and e 1 and e 2 E w=1. Then (H(G; e 1, e 2 ), w) is nonhamiltonian coverable if and only if e 1 and e 2 are not eventually adjacent. Let G, w and {e 1, e 2 } be defined as in Theorem 3.5. We shall prove Lemma 3.6 for the necessity and Lemma 3.8 for the sufficency of Theorem

10 Lemma 3.6 If (H(G; e 1, e 2 ), w) is nonhamiltonian coverable, then e 1 and e 2 are not eventually adjacent. Proof. We argue by contradiction and assume that e 1 and e 2 are eventually adjacent. By Lemma 3.4, H = H(G; e 1, e 2 ) S. Let M = E w=2 (G) and M = E w=2 (H). Then M = M {e }, where e is defined in Definition 2.1. Let T be a 3-cut of H. If e T, then T can be viewed as a 3-cut of G, and so by Lemma 2.3, T M = T M = 1. If e T, then since G is 3-edge-connected, T e cannot be an edge-cut of G. Therefore, T must consist of the three edges incident with x 1 or with x 2, and so T M = 1 also. By Lemma 2.3, w is a Hamilton weight of H(G; e 1, e 2 ), contrary to the assumption that (H(G; e 1, e 2 ), w) is nonhamiltonian coverable. Definition 3.7 Let w be a (1, 2)-eulerian weight of a cubic graph G. A 1-factorization M of G is w-matched if E w=2 M, and is w-unmatched if E w=2 M. If we are to prove that a cubic graph G with a (1, 2)-eulerian weight w is nonhamiltonian coverable, by Lemma 2.15, we need only to find a w-unmatched 1-factorization. Lemma 3.8 Let G, w, and {e 1, e 2 } be defined as in Theorem 3.5. If the edges e 1 and e 2 are not eventually adjacent, then H(G; e 1, e 2 ) has a w-unmatched 1-factorization. We shall prove Lemma 3.8 by induction. needed to reduce the order of the graph. The following lemmas and notions will be Lemma 3.9 Let G be a cubic graph and w be a (1, 2)-eulerian weight of G such that E w=2 is a perfect matching of G, and let e 1 and e 2 E w=1. Assume that G has a triangle C = abca which contains e 2 = ab but not e 1 and w(ac) = 2. Let e 2 = cc be the edge incident with c but not in the triangle C. Let G be the graph obtained from G by contracting C to a vertex and w be the restriction of w on G. (See Figure 2.) Then H(G; e 1, e 2 ) has a w-unmatched 1-factorization if H(G ; e 1, e 2 ) has a w -unmatched 1-factorization. (Note that e 2 and e 2 are contained in the same member of any 1 factorization of G.) c c e 2 c z a b e 2 b a G b G Figure 2: Oeration defined in Lemma

11 Proof. Let H = H(G; e 1, e 2 ). The new vertex created by the contraction is denoted by z, and the vertex inserted into e 2 is denoted by x 2. Let M = {M 1, M 2, M 3 } be a w - unmatched 1-factorization of G. Without loss of generality and by symmetry, we may assume that zx 2 M 1 and zb M 3. Then za M 2, and either x 2 x 1 M 2 or x 2 x 1 M 3. Case 1. x 2 x 1 M 3. Then x 2 c M 2. Let M = {M 1, M 2, M 3 } be a 1-factorization of H such that M 1 = [M 1 \ {zx 2}] {x 2 a, cb}, M 2 = [M 2 \ {za, x 2c }] {c c, a a, x 2 b}, M 3 = [M 3 \ {zb, x 2x 1 }] {x 1 x 2, b b, ca}. Since M is w-unmatched, (M 1 E w =2(G )) (M 2 E w =2(G )). (M 1 E w=2 (H)) (M 2 E w=2 (H)) also, and so M is w-unmatched. Case 2. x 2 x 1 M 2. Then x 2 c M 3. Let M = {M 1, M 2, M 3 } be a 1-factorization of H such that It follows that M 1 = [M 1 \ {zx 2}] {ac, bx 2 }, M 2 = [M 2 \ {za, x 2x 1 }] {a a, cb, x 1 x 2 }, M 3 = [M 3 \ {zb, x 2c }] {c c, b b, x 2 a}. Obviously M is w-unmatched since M is w -unmatched. Lemma 3.10 If e 1 and e 2 are contained in the same member of the 1 factorization of a graph G S, then e 1 and e 2 are not eventually adjacent. Proof. This is trivial if G {K2 3, K 4}. This lemma can be proved by induction on V (G), by applying Y -operation to every triangle of G containing neiter e 1 nor e 2, and by Lemma 3.3. Proof of Lemma 3.8. Assume that e 1 and e 2 are not eventually adjacentand we shall argue by induction. Let H = H(G; e 1, e 2 ). The lemma is obvious for graphs with at most 4 vertices: Any two edges in K2 3 are adjacent; and if e 1, e 2 E(K 4 ) are not adjacent, then H(K 4 ; e 1, e 2 ) = K 3,3, which has a w-unmatched 1-factorization. If G has a triangle C 0 such that E(C 0 ) {e 1, e 2 } =, then obtain G 1 S from G by contracting C 0 into a single vertex, and let w 1 denote the restriction of w to E(G 1 ) = E(G) \ E(C 0 ). By induction, H(G 1 ; e 1, e 2 ) has a w 1 -unmatched 1-factorization M 1. It is straight forward to check that M 1 would induce a w-unmatched 1-factorization of H(G; e 1, e 2 ). 11

12 Therefore every triangle of G must contain exactly one edge in {e 1, e 2 }, and so G has exactly two triangles C 1 and C 2 where e i E(C i ) for i = 1, 2. By Lemma 2.6, G = Γ(n, φ) for some n and φ. Assume that C 2 = x 0 y 0 z 0 x 0. If the range of φ has at least two elements, then we may assume, without loss of generality, that there exist integers n 1 > 0 and n 2 > 0 such that φ(i) = z for 1 i n 1, φ(i) = x for n i n 2 and φ(n 2 + 1) x. Apply Operation 1 to G to get G by contracting C 2. Then the edge e 2 must then be one of the three edges in G \ E(C 2) incident with a vertex of V (C 2 ). Since n 1 > 0 and n 2 > 0, either e 2 is in the newly formed triangle, whence e 2 and e 1 are in two distinct triangles of G ; or e 2 and e 1 are in two distinct triangles of the resulting graph obtained from G by subsequently applying Y -operations to contract triangles containing edges of the form x i y i with 1 i n 1 1. By Lemma 3.3, e 2 and e 1 are not eventually adjacent, and so by induction and by Lemma 3.9, H(G; e 1, e 2 ) has a w-unmatched 1-factorization. Therefore the range of φ must have only one element, and so G = Λ(n) for some n 3, by Lemma 2.6(4). We shall use the notations in Lemma 2.6 for Λ(n), and so V (Λ(n)) = {v 1, v 2,, v 2n }. Without loss of generality, we may assume that w(v 1 v 2 ) = 2, C 1 = v 1 v 2 v 2n v 1 and C 2 = v n v n+1 v n+2 v n. (See Figure 4.) By Lemma 3.10, we may assume that e 1 and e 2 are in different members of the 1- factorization of G. Thus we shall define M 1 as follows. If n is even, then let M 1 = {e, v 1 v n+1, v n v n+2 } {v 2i v 2i+1 1 i n 2 } {v 2j 1 v 2j n + 4 j n}. 2 2 If n is odd, then let M 1 = {e, v 1 v n+1 } {v 2i v 2i+1 1 i n 1 } {v 2j 1 v 2j n + 3 j n}. 2 2 In either case, H(Λ(n); e 1, e 2 ) \ M 1 is a disjoint union of even circuits, and M 1 contains e and edges with weight 1. Therefore H(G; e 1, e 2 ) has a w-unmatched 1-factorization. 4 The Proofs of the Main Results The sufficiency of Theorem 1.1 follows trivially from Lemma 2.3. We shall argue by contradiction to prove the necessity. Choose (G, w) to be a counterexample to Theorem 1.1 with V (G) minimized. Thus (G, w) is Hamilton but G S. 12

13 I. By the minimality of G and by Lemma 2.7, G is 3-connected and cyclically 4-edgeconnected. In fact, if G = G 1 (v1,v 2 ) G 2 for two cubic graphs G 1 and G 2, then both (G 1, w) and (G 2, w) are Hamilton. By the minimality of G, G 1, G 2 S, and so by Lemma 2.7, G S also, a contradiction. II. By Lemma 2.14, the subgraph of G induced by E w=1 is a Hamilton circuit C of G. By Lemmas 2.12 and 2.18, it follows that G has an edge e E w=2 which is an even pace chord of C; and (G \ {e}, w) has a faithful circuit cover C = {C 1,, C 2t } with C = 2t maximized, where C is a circuit chain joining the end vertices x and y of e. That is, x C 1, y C 2t and C i C j if and only if i j = ±1. III. Let J i = C i C i+1 (i = 1,, t 1). Let w i be the weight of J i induced by the family of circuits {C i, C i+1 }. For a technical reason, let {x} = C 0 and {y} = C 2t+1. We claim that the underlying graph J i S, for each i {1,, 2t 1}. Since G is a smallest counterexample to the theorem, it suffices to show i {1,, 2t 1}, J i is 3-connected, admits a Hamilton weight and has no Petersen-minor. By the maximality of C, the induced weight w i is a Hamilton weigh for each J i. As each J i is a subgraph of G, J i contains no subdivision of the Petersen graph. It remains to show that J i is 3-connected. Since J i is cubic, it suffices to show that J i has no 2-edge-cuts. By contradiction, suppose that T is a 2-edge-cut of J i for some i. Let R 1 and R 2 be two components of J i \ T and let {D 1,, D s } be a circuit decomposition of E wi =1 where, by Lemma 2.10, each D µ is contained in either R 1 or R 2. If C i 1 C i+2 intersects with only one of {D 1,, D s }, then then T is a 2-edge-cut of G, contrary to Claim I. Thus, there are two distinct circuits, say D 1 and D 2, of {D 1,, D s } such that D 1 C i 1 and D 2 C i+2. By Lemma 2.10 again, {C i D 1, C i+1 D 1 } is also a Hamilton cover of (J i, w i ). Thus, C 1,, C i 1, C i+1 D 1, C i+2,, C t is a circuit chain of G \ {e 0 } joining the endvertices of e 0 with a circuit C i D 1 ( ) excluded because (C i+1 D 1 ) C i 1 D 1 C i 1, and (C i+1 D 1 ) C i+2 D 2 C i+2. This contradicts Lemma Thus, J i must be 3-connected and so Claim III is proved. For each i, call a subdivided edge L of J i an attachment of C i 1 (or C i+2 ) if C i 1 L (or C i+2 L, respectively). 13

14 IV. Suppose that some J i, where 1 i 2t, has attachment edges e 1 (E(J i ) \ E(C i+1 )) of C i 1 and e 2 (E(J i ) E(C i )) of C i+2, respectively. Then we claim that e 1 and e 2 are eventually adjacent in J i. By contradiction, assume that e 1 and e 2 are no eventually adjacent. By Claim III and by Theorem 3.5, let D 1 and D 2 be the circuits of a nonhamiltonian cover C i of H(J i ; e 1, e 2 ) containing the new edge e. Let G be the subgraph of G induced by edges contained in all circuits of C\{C i, C i+1 }, the paths D 1 \{e } and D 2 \{e }, and the edge e 0. Let w be the weight of G induced by the family of subgraphs [C \ {C i, C i+1 }] {D 1 \ {e }, D 2 \ {e }, {e 0 }, {e 0 }}. The new weight w is eulerian, and the new graph G is bridgeless and Petersen minor free. By Theorem 2.9, (G, w ) has a faithful circuit cover F. Therefore, F [C i \ {D 1, D 2 }] is a faithful Hamilton cover of (G, w). This contradicts to the assumption that w is a Hamilton weight of G since C i \ {D 1, D 2 }. This proves Claim IV. V. We claim that t 2. That is, the circuit chain C is of length at least 4. Assume that t = 1. That is, the circuit chain C = {C 1, C 2 } is of length two. By Claim III, J 1 = C 1 C 2 S. Let e 1, e 2 be the subdivided edges of C 1 C 2 containing the endvertices of e. Thus, G = H(C 1 C 2 ; e 1, e 2 ). By Theorem 3.5, e 1 and e 2 must be eventually adjacent since w is a Hamilton weight of G. By Lemma 3.4, G = H(C 1 C 2 ; e 1, e 2 ) S, contrary to the assumption that (G, w) is a counterexample, and so Claim V holds. VI. If C i+2 has only one attachment in J i when i + 1 < 2t (or, if C i 1 has only one attachment in J i when i > 1), then G has a cyclical 3-edge-cut consisting of the edge e 0 and two edges of C i+1 (or two edges of C i, respectively), contrary to Claim I. Thus each of C i 1 (when i > 1) and C i+2 (when i + 1 < 2t) has at least two attachments in J i. Fix an i {2,, 2t 2}. Assume that J i S \ {K 3 2, K 4}. Then we can find a subdivided edge e 1 of J i \ C i+1 C i which is an attachment of C i 1, and a subdivided edge e 2 of J i \ C i C i+1 which is an attachment of C i+2. Since G is cyclically 4-edge-connected, and since J i S \ {K 3 2, K 4}, each triangle of J i must contain an edge of attachment of either C i 1 or C i+2. By Lemma 2.6, J i has at least 2 disjoint triangles. Thus by Lemma 3.3, one can choose the edges e 1 and e 2 of J i S such that they are not eventually adjacent, contrary to Claim IV. Thus each J i is either K 4 or K2 3 for each i = 2,, 2t 2. However, by Claim VI, J i K2 3. Therefore, the next claim obtains. VII. J i = K 4, for each i with 2 i 2t 2, and J i S \ {K2 3 } for each i {1, 2t 1}. 14

15 VIII. For any 1 i 2t 2, if J i = K 4, then J i+1 K 4. By contradiction and without loss of generality, we assume that both J i = C i C i+1 = K 4 and J i+1 = C i+1 C i+2 = K 4. Consider the subgraph C i C i+1 C i+2, depicted in Figure 3. Since t 2, we may assume, without loss of generality, that i + 2 2t 1. Thus i + 3 2t. We shall use the notation in Figure 3 to proceed the proof for Claim VIII. By Claim VI, 12 is an attachment of C i 1 and both 56 and 78 are attachments of C i+3. Note that the circuit is adjacent to both C i 1 and C i+3. Thus, by replacing C i, C i+1 and C i+2 of the circuit chain C with the circuits and , we obtain a new faithful circuit cover of (G \ {e}) which is not a circuit chain joining x and y, contrary to Lemma This proves Claim VIII = C i C i+1 C i+2 C i C i+1 C i+2 Figure 3: C i C i+1 C i+2 where C i C i+1 = K4 and C i+1 C i+2 = K4 By Claim V, t 2. If t 3, then by Claim VII, both J 2 = K 4 and J 3 = K 4, contrary to Claim VIII. Therefore t = 2, and so by Claims VII and VIII, (G \ {e 0 }) has a faithful cover {C 1, C 2, C 3, C 4 } such that J 1, J 3 S \ {K2 3, K 4}, and J 2 = K4. IX. J 1 has exactly two edges of attachments of C 3, since J 2 = K 4. It follows that C 3 E(C 2 ) consists of exactly two maximal subdivided edges in J 2. X. By Claim I, G is cyclically 4-edge-connected. Therefore each triangle of J 1 must contain an edge of attachment of C 0 or of C 3. XI. Let e 0 be the subdivided edge of C 1 containing {x} = C 0. By Claims IV and X, we conclude that e 0, the only edge of attachment of C 0 in J 1 is not in any triangle of J 1, that J 1 has exactly two triangles T 1 and T 2, and that J 1 has two edges e 1, e 2 of attachments of C 3 with e 1 E(T 1 ) and e 2 E(T 2 ). By Claim IV again, e 0 is adjacent to both e 1 and e 2. 15

16 By Lemma 2.6(4) and by the assumption that J 1 {K2 3, K 4}, J 1 = Λ(n) for some n 0 (Figure 4). v 1 v a 1 b v 12 v 6 v 12 v 6 v 11 v 5 v 11 v 5 v 10 v 4 v 10 v 4 v 9 v 3 v 9 v 3 v 8 v 2 v 7 (a) Γ(6) v 8 v 2 c d v 7 (b) Γ(6) with attachments ab and cd Figure 4 We shall use the notations in Lemma 2.6 for Λ(n), and so V (Λ(n)) = {v 1, v 2,, v 2n }. Without loss of generality, we assume that T 1 = v 1 v 2 v 2n v 1 and T 2 = v n v n+1 v n+2 v n, and that C 1 = v n+1 v 1 v 2 v 2n v 2n 1 v 3 v n+2 v n+1 when n is even, or C 1 = v n+1 v 1 v 2 v 2n v 2n 1 v 3 v n v n+1 when n is odd; C 2 = v 1 v 2 v 3 v 2n v 1 ; and the vertex x is contained in the subdivided edge v 1 v n+1. By Claim VIII, J 2 = K 4 and by Claim IX, J 1 has two edges of attachment of C 3 which must be in distinct triangles of J 1. Thus we may assume that these edges of attachment of C 3 in J 1 are the subdivided edges of J 1 : v 1 bav 2n in T 1 with ab E(C 2 ) E(C 3 ); and v n cdv n+1 in T 2 (if n is even), or v n+1 cdv n+2 in T 2 (if n is odd), with cd E(C 2 ) E(C 3 ). Since J 2 = K 4, the subdivided edges ac and bd are in E(C 3 ), and so C 2 {ac, bd} = K 4. Note that the two edges of attachment of C 4 in J 2 must then be the subdivided edges ac and bd of C 3. Let N = C 1 C 2 C 3. We shall construct another faithful circuit cover {C 1, C 2, C 3 } of N with respect to the weight induced by {C 1, C 2, C 3 }. We shall adopt the following notation in the construction: If C = z 1 z 2 z m z 1 denotes a circuit, then z i Cz j denotes the section z i z i+1 z j 1 z j of C. If n is even, then let C 1 = v 1 v 2 v 3 C 1 v n+1 v 1, C 2 = bav 2n v 2n 1 v 3 C 2 v n cdb, C 3 = v 1 bacdv n+1 C 2 v 2n v 2 v 1. 16

17 If n is odd, then let C 1 = v 1 v 2 v 3 C 1 v n+1 v 1, C 2 = bav 2n v 2n 1 v 3 C 2 v n+1 cdb, C 3 = bacdv n+2 C 2 v 2n v 2 v 1 b. In either case, {C 1, C 2, C 4} forms a circuit chain joining the end vertices x and y of e with C 3 excluded, contrary to Lemma This completes the proof. In the proof for Corollary 1.2, we shall use the same notation in Section 1 for the definition of G 1 (e1,e 2 ) G 2. For each i {1, 2}, let (G i, w i ) be a cubic graph with a (1,2)- weight w i and let e i E(G i ) be an edge with w i (e i ) = 2. Let G = G 1 (e1,e 2 ) G 2 and let w : E(G ) {1, 2} be such that w (e) = w i (e) if e E(G ) \ {e x, e y }, and w (e x ) = w (e y ) = 2. Part(i) of the following lemma follows from the facts that {e x, e y } E w =2(G ) and that e i E wi =2(G i ), with 1 i 2; and Part(ii) of it follows from the definition of G and from the fact that the Petersen graph is 3-connected. Lemma 4.1 With G defined as above, each of the following holds: (i) w is a Hamilton weight of G if and only if both w 1 is a Hamilton weight of G 1 and w 2 is a Hamilton weight of G 2. (ii) G does not have a Petersen minor if and only if neither G 1 nor G 2 has a Petersen minor. Corollary 4.2 Let (G, w ) be a cubic graph with a Hamilton weight w. Then either G is 3-connected, or there exist cubic graphs (G 1, w 1 ) and (G 2, w 2 ) with Hamilton weights, such that, for some edges e 1 E(G 1 ) and e 2 E(G 2 ) with w 1 (e 1 ) = w 2 (e 2 ) = 2, G = G 1 (e1,e 2 ) G 2. Proof. Since G admits a Hamilton weight, G is 2-connected. Suppose that G is not 3- connected. Then G has an edge 2-cut T = {e x, e y } (say). By Lemma 2.10, T E w =2(G ). Let G 1 and G 2 denote the two components of G \ T and assume that e x = x 1 x 2 and e y = y 1 y 2, where x 1, y 1 V (G 1 ) and x 2, y 2 V (G 2 ). Since G is 2-connected and cubic, x 1 y 1 and x 2 y 2. Fix an i {1, 2}. Let G i be the graph obtained from G i by adding a new edge e i = x i y i. Let w i : E(G i ) {1, 2} be such that w i (e) = w (e) if e {e 1, e 2 }, and w 1 (e 1 ) = w 2 (e 2 ) = 2. Then G = G 1 (e1,e 2 ) G 2, and by Lemma 4.1, both (G 1, w 1 ) and (G 2, w 2 ) are cubic graphs with Hamilton weights. We are now ready to prove Corollary 1.2. It suffices to prove the necessity. Let G be a cubic graph without a Petersen minor. If G is 3-connected, then by Theorem 1.1, G S. Assume that G has an edge 2-cut. Then by Corollary 4.2, there exist cubic graphs G 1 and G 2, each of which admits a Hamilton weight, such that, for some edges e 1 E(G 1 ) and e 2 E(G 2 ), G = G 1 (e1,e 2 ) G 2. Note that since both G 1 and G 2 are cubic graphs, V (G i ) V (G) 2 for each i {1, 2}. Thus Corollary 1.2 obtains by applying induction to both G 1 and G 2. 17

18 5 Critical contra pairs and Petersen minor Definition 5.1 A cubic graph G is called a permutation graph if G has a 2-factor F which is the union of two chordless circuits. Definition 5.2 An ordered pair (G, w) is called a contra pair if w is an admissible eulerian weight of G and (G, w) has no faithful cycle cover. It is proved in [1] and [2], every minimal contra pair must be a permutation graph G with E w=1 as its 2-factor which is the union of two chordless circuits, and E w=2 as the set of edges joining the components of E w=1. By applying a theorem of Ellingham [3], it, then, proves the existence of a Petersen-minor in such a permutation graph. The main result in this section proves that under certain minimal condition, the Petersen-minors appear almost everywhere in such a permutation graph. Definition 5.3 Let w be an admissible (1, 2)-eulerian weight of a cubic graph G. Let C be a circuit of G and let w C be the eulerian weight of G such that { w(e) if e E(C) w C (e) = w(e) 1 if e E(C). (G, w) is a critical contra pair if for each circuit C of G, the eulerian weight w C is not admissible. Let P 10 be the Petersen graph, and M be a perfect matching of P 10. Define a (1, 2)- eulerian weight w 10 of P 10 as follows. w 10 (e) = { 2 if e M 1 otherwise. (see Figure 5.) One can prove that (P 10, w 10 ) does not have a faithful cycle cover. And by Theorem 2.9, (P 10, w 10 ) is the smallest contra pair. Figure 5: The Petersen graph P 10 18

19 Conjecture 5.4 (Jackson [12]) Let G be a cyclically 5-edge-connected cubic graph and w be a (1,2)-eulerian weight of G. If (G, w) has no faithful cover, then G = P 10. Conjecture 5.5 (Goddyn [7], or see [8]) (P 10, w 10 ) is the only 3-connected and cyclically 4-edge-connected critical contra pair. Conjecture 5.6 (Fleischner and Jackson, Conjecture 12 in [4]) Let G be a permutation graph such that M is a perfect matching of G and G\M is the union of two chordless circuits C 1 and C 2. If (G, w) is a critical contra pair with E w=2 = M and E w=1 = E(C 1 ) E(C 2 ), then (G, w) = (P 10, w 10 ). The main result of this part is a byproduct of the proof of Theorem 1.1 and is an approach to the above conjectures. Theorem 5.7 Let (G, w) be described in Conjecture 5.6. Let e E w=2 = M. A. Then (G \ {e}, w) is faithful coverable, furthermore, every faithful cover of (G \ {e}, w) is a circuit chain joining the endvertices of e. B. Choose a faithful cover F = {C 1,, C s } with F = s as large as possible. If s 4, then C i C i+1 C i+2, for each i {1,, s 2}, contains a Petersen minor. Actually, we are able to prove a slightly stronger result. Theorem 5.8 Let (G, w) be described in Conjecture 5.6. Let e E w=2 = M. A. Then (G \ {e}, w) is faithful coverable, furthermore, every faithful cover of (G \ {e}, w) is a circuit chain joining the endvertices of e. B. Choose a faithful cover F = {C 1,, C s } with F = s as large as possible. (1) Then s 3; (2) If s 4, then, for each i {2,, s 2}, either C i 1 C i or C i+1 C i contains a Petersen minor; (3) If s 4 and C i 1 C i does not contain a Petersen minor, for some i {3,, s 1}, then C i 1 C i = K 4 ; Lemma 5.9 Let G be a permutation graph. Then G is cyclically 4-edge-connected if V (G) 8. Proof. Let M be a perfect matching of G and G \ M is the union of two chordless circuits C 1 and C 2. Since C 1 C 2 is a 2-factor of G, each cyclic edge-cut T of G must contain an even number of edges of C 1 C 2. Assume that a permutation graph G is not cyclically 4-edge-connected. Let T be a cyclic edge-cut of size at most 3. If T (C 1 C 2 ) = 2, then one of {C 1, C 2 }, say C 1, does not intersect with one component of G \ T. Thus, 19

20 some edge of M must be a chord of C 2, and therefore, G is not a permutation graph. If T (C 1 C 2 ) = 0, then T M. It is not hard to see that C 1 (as well as C 2 ) is a spanning subgraph of a component of G \ T. Since each component of G \ T is a chordless circuit, no component of G \ T contains any edge of M. This implies that M = T and therefore V (G) = 2 T = 2 M 6. Proof of Theorem 5.8. Since (G, w) is a contra pair, G is not edge-3-colorable. Therefore, the length of the circuits C 1, C 2 must be odd and is at least 5. Furthermore, for each edge e M = E w=2, G \ {e} is edge 3-colorable. Hence, (G \ {e}, w) has a faithful circuit cover. Since (G, w) is a critical contra pair, every faithful cover of (G \ {e}, w) is a circuit chain. The length of the circuit chain must of length at least two since M \ {e} is not empty. (1) of the Theorem 5.8 can be proved easily as follows. Color the edges of G \ {e} with three colors such that M = E w=2 is colored red, edges of C 1 and C 2 in the underlying graph G \ {e} are alternatively colored with blue and yellow. Alternating colors of C 1 if necessary so that the pair of subdivided edges of G \ {e} (containing the endvertices of e) are colored same. Then a circuit chain of (G \ {e}, w) (with each element colored by two colors) joining the endvertices of e must be of odd length. By Lemma 5.9, G is a cyclically 4-edge-connected permutation graph. Let F = {C 1,, C s } be a faithful cover of (G \ {e 0 }, w) for some e 0 = xy E w=2 with F as large as possible. Here C i C j if and only if i = j ± 1 since F is a circuit chain. For a technical reason, let {x} = C 0 and {y} = C t+1. Let J i = C i C i+1 (i = 1,, t 1). Let w i = w {C1,C i+1 } be the weight of J i induced by the family of circuits {C i, C i+1 }. Since F is maximum, each J i admits a Hamilton weight w i = w {Ci,C i+1 }. By Theorem 1.1, either J i contains a Petersen minor or J i S. The rest of the proof is quite similar to that in the proof of Theorem 1.1, and so we only present an outline of it. Similar to Claim VI in Section 4, we have that each of C i 1 (when i > 1) and C i+2 (when i+1 < s) has at least two attachments in J i. Therefore, J i K2 3 for each i. Similar to Claim VII in Section 4, we have that for each i with 2 i s 2, if J i does not contain the Petersen minor, then J i K 4. This proves (3) of Theorem 5.8. Similar to Claim VIII in Section 4, we have that 20

21 for each i with 1 i s 2, if J i K 4 then both J i 1 (when i 2) and J i+1 (when i s 2) must contain a Petersen minor. This proves (2) of Theorem 5.8. Remark It was announced by Tom Fowler at the 28th Southeastern International Conference on Combinatorics, Graph Theory And Computing (Florida Atlantic University, Boca Raton, Florida, March, 1997), that Conjecture B of Fiorini and Wilson on unique edge- 3-coloring for planar graphs has been solved by Robin Thomas and Tom Fowler with an approach similar to that for the 4-color theorem in [14]. References [1] B. Alspach and C.-Q. Zhang, Cycle covers of cubic multigraphs, Discrete Math., 111, (1993) p [2] B. Alspach, L. A. Goddyn and C.-Q. Zhang, Graphs with the circuit cover property, Trans. Amer. Math. Soc., Vol. 344, No. 1, (1994) p [3] M. N. Ellingham, Petersen subdivisions in some regular graphs, Congr. Numer., Vol. 44, Nov. (1984), p [4] H. Fleischner, Some blood, sweat, but no tears in eulerian graph theory, Congr. Numer., Vol. 63, Nov. (1988), p [5] S. Fiorini and R. J. Wilson, Edge colourings of graphs, Research Notes in Mathematics No. 16, Piman (1977). [6] S. Fiorini and R. J. Wilson, Edge colourings of graphs. in Selected Topics in Graph Theory, (L. Beineke and R. J. Wilson eds.), Academic Press, London, (1978) p [7] L. A. Goddyn, Cycle double covers current status and new approaches, Contributed lecture at Cycle Double Cover Conjecture Workshop, IINFORM, Vienna, January [8] L. A. Goddyn, J. van den Heuvel and S. McGuinness, Removable Circuits in Multigraphs, (preprint). [9] J. L. Goldwasser and C. Q. Zhang, On minimal counterexamples to a conjecture about unique edge-3-c loring, Congr. Numer., (to appear). 21

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