A cycle decomposition conjecture for Eulerian graphs

A classic theorem of Veblen states that a connected graph G has a cycle decomposition if and only if G is Eulerian. The number of odd cycles in a cycle decomposition of an Eulerian graph G is therefore even if and only if G has even size. It is conjectured that if the minimum number of odd cycles in a cycle decomposition of an Eulerian graph G with m edges is a and the maximum number of odd cycles in a cycle decomposition is c, then for every integer b such that a ≤ b ≤ c and b and m are of the same parity, then there is a cycle decomposition of G with exactly b odd cycles. This conjecture is verified for small powers of cycles and Eulerian complete tripartite graphs. ∗ Also at Department of Mathematics, Illinois State University, Normal, IL 61790-4520, U.S.A. G. CHARTRAND ET AL. /AUSTRALAS. J. COMBIN. 58 (1) (2014), 48–59 49 1 A Circuit Decomposition Problem It is well-known that if G is a connected graph containing 2k odd vertices for some positive integer k, then G can be decomposed into k open trails but no fewer. In 1973, the following [8] was proved. Theorem 1.1 If G is a connected graph containing 2k odd vertices for some positive integer k, then G can be decomposed into k open trails, at most one of which has odd length. A generalization of Theorem 1.1 was established in [4]. Theorem 1.2 Let G be a connected graph of size m containing 2k odd vertices (k ≥ 1). Among all decompositions of G into k open trails, let s be the maximum number of such trails of odd length. (a) If m is even, then s is even and for every even integer a such that 0 ≤ a ≤ s, there exists a decomposition of G into k open trails, exactly a of which have odd length. (b) If m is odd, then s is odd and for every odd integer b such that 1 ≤ b ≤ s, there exists a decomposition of G into k open trails, exactly b of which have odd length. The distance between two subgraphs F and H in a connected graph G is d(F,H) = min{d(u, v) : u ∈ V (F ), v ∈ V (H)}. Theorem 1.3 For an Eulerian graph G of size m, let s be the maximum number of circuits of odd length in a circuit decomposition of G. (a) If m is even, then s is even and for every even integer a such that 0 ≤ a ≤ s, there exists a circuit decomposition of G, exactly a of which have odd length. (b) If m is odd, then s is odd and for every odd integer b such that 1 ≤ b ≤ s, there exists a circuit decomposition of G, exactly b of which have odd length. Proof. We only verify (a) because the proof of (b) is similar. Since the size of G is even, s is even. If s = 0, then the result is true trivially. Thus we may assume that s ≥ 2. It suffices to show that there exists a circuit decomposition of G, exactly s − 2 of which have odd length. Among all circuit decompositions of G, consider those circuit decompositions containing exactly s circuits of odd length; and, among those, consider one, say D = {C1, C2, . . . , Ck} for some positive integer k, where the distance between some pair Ci, Cj of circuits of odd length is minimum. We claim that this minimum distance is 0. Assume that this is not the case. Suppose that P G. CHARTRAND ET AL. /AUSTRALAS. J. COMBIN. 58 (1) (2014), 48–59 50 is a path of minimum length connecting a vertex wi in Ci and a vertex wj in Cj , and let wix be the edge of P incident with wi (where it is possible that x = wj). Then wix belongs to a circuit Cp among C1, C2, . . . , Ck. Necessarily, Cp has even length, for otherwise, the distance between Ci and Cp is 0, producing a contradiction. Since Ci and Cp have the vertex wi in common, Ci and Cp may be replaced by the circuit C ′ consisting of Ci and Cp (that is, E(C ′) = E(Ci)∪E(Cp)) and C ′ has odd length. However then, the circuit decomposition D′ = ({C1, C2, . . . , Ck} − {Ci, Cp}) ∪ {C ′} has exactly s circuits of odd length and the distance between Cj and C ′ in D′ is smaller than the distance between Ci and Cj in D, which contradicts the defining property of D. Thus, as claimed, the distance between Ci and Cj is 0 and so Ci and Cj have a vertex in common. Hence the circuit C ∗ consisting of Ci and Cj has even length. Then ({C1, C2, . . . , Ck} − {Ci, Cj}) ∪ {C∗} is a circuit decomposition of G, exactly s− 2 of which have odd length. 2 The Eulerian Cycle Decomposition Conjecture The earliest and a major influential book on topology was written by Veblen [17] in 1922 and titled Analysis Situs, with a second edition in 1931. The first chapter of this book was titled Linear Graphs and dealt with graph theory. In fact, both editions preceded the first book entirely devoted to graph theory, written by König [14] in 1936. In 1736 Euler [9] wrote a paper containing a solution of the famous Königsberg Bridge Problem. This paper essentially contained a characterization of Eulerian graphs as well, although the proof was only completed in 1873 in a paper by Hierholzer [12]. In 1912 Veblen [16] himself obtained a characterization of Eulerian graphs. Theorem 2.1 (Veblen’s Theorem) A nontrivial connected graph G is Eulerian if and only if G has a decomposition into cycles. When it comes to cycle decompositions, the Eulerian graphs that have received the most attention are the complete graphs of odd order and, to a lesser degree, the complete graphs of even order in which (the edges of) a 1-factor has been removed. In 1847, Kirkman [13] proved that the complete graph Kn, where n ≥ 3 is odd, can be decomposed into 3-cycles if and only if 3 | (n 2 ) . At the other extreme, in 1890 Walecki (see [2]) proved that the complete graph Kn, where n ≥ 3 is odd, can always be decomposed into n-cycles. Consequently, when n ≥ 3 is an odd integer, the complete graph Kn can be decomposed into m-cycles for m = 3 or m = n if and only if m | (n 2 ) . In 2001 Alspach and Gavlas [3] proved for every odd integer n ≥ 3 and odd integer m with 3 < m < n that Kn can be decomposed into m-cycles if and only if m | (n 2 ) . In addition, they proved that for every even integer n ≥ 4 and even integer m with 3 < m < n and for a 1-factor I of Kn, the graph Kn − I can be decomposed into m-cycles if and only if m | (n − 2n)/2. In 2002, Šajna [15] proved the remaining cases for m-cycle decompositions of Kn and Kn − I, namely G. CHARTRAND ET AL. /AUSTRALAS. J. COMBIN. 58 (1) (2014), 48–59 51 the cases when m and n are of opposite parity. These results verify special cases of a conjecture made by Alspach [1] in 1981. Alspach’s Conjecture Suppose that n ≥ 3 is an odd integer and that m1, m2, . . . , mt are integers such that 3 ≤ mi ≤ n for each i (1 ≤ i ≤ t) and m1+m2+· · ·+mt = ( n 2 ) . Then Kn can be decomposed into the cycles Cm1 , Cm2, . . . , Cmt. Furthermore, for every even integer m ≥ 4 and integers m1, m2, . . . , mt such that 3 ≤ mi ≤ n for each i (1 ≤ i ≤ t) with m1 + m2 + · · · + mt = (n − 2n)/2, there is a decomposition of Kn − I for a 1-factor I of Kn into the cycles Cm1 , Cm2, . . . , Cmt. Following many years of attempting to establish Alspach’s Conjecture by many mathematicians, the conjecture was verified in its entirety by Bryant, Horsley and Pettersson [6] in 2012. We now state another conjecture involving cycle decompositions of Eulerian graphs. The Eulerian Cycle Decomposition Conjecture (ECDC) Let G be an Eulerian graph of size m, where a is the minimum number of odd cycles in a cycle decomposition of G and c is the maximum number of odd cycles in a cycle decomposition of G. For every integer b such that a ≤ b ≤ c and b and m are of the same parity, there exists a cycle decomposition of G containing exactly b odd cycles. In the case of the complete graphs of odd order or complete graphs of even order in which a 1-factor has been removed, the maximum number of odd cycles in a cycle decomposition of each such graph is given below. This follows from results of Kirkman [13], Guy [10] and Heinrich, Horák and Rosa [11]. Corollary 2.2 (a) For an odd integer n ≥ 3, the maximum number s of odd cycles in a cycle decomposition of Kn is s = { n(n−1) 6 if n ≡ 1, 3 (mod 6) n(n−1)−8 6 if n ≡ 5 (mod 6). (b) For an even integer n ≥ 4 and a 1-factor I of Kn, the maximum number s of odd cycles in a cycle decomposition of Kn − I is s = { n(n−2) 6 if n ≡ 0, 2 (mod 6) n(n−2)−8 6 if n ≡ 4 (mod 6). For complete graphsKn of odd order n ≥ 3 and graphsKn−I where n ≥ 4 is even and I is a 1-factor of Kn, the ECDC is then a special case of Alspach’s Conjecture and therefore is satisfied for these two classes of graphs. 3 The ECDC and Small Powers of Cycles In a cycle decomposition of an Eulerian graph G, the number of odd cycles in the decomposition and the size of G are of the same parity. One class of Eulerian graphs G. CHARTRAND ET AL. /AUSTRALAS. J. COMBIN. 58 (1) (2014), 48–59 52 consists of the squares C n of cycles Cn where n ≥ 5, and more generally the kth power C n of Cn for k ≤ n/2 , which is a special class of circulant graphs. For each integer n ≥ 3 and integers n1, n2, . . . , nk (k ≥ 1) such that 1 ≤ n1 < n2 < . . . < nk ≤ n/2 , the circulant graph 〈{n1, n2, . . . , nk}〉n is that graph with n vertices v1, v2, . . . , vn such that vi is adjacent to vi±nj (mod n) for each j with 1 ≤ j ≤ k. The integers ni (1 ≤ i ≤ k) are called the jump sizes of the circulant. The circulant graph 〈{1, 2, . . . , k}〉n is the kth power of Cn and is denoted by C n and in particular, if k = 1, then 〈{1}〉n = Cn. The circulant 〈{n1, n2, . . . , nk}〉n is 2k-regular if nk < n/2 and (2k − 1)-regular if nk = n/2 where then n is even. Thus circulant graphs are symmetric classes of regular graphs. Let G be an Eulerian graph of order n and size m. For a sequence m1, m2, . . . , mt of positive integers, an (m1, m2, . . . , mt)-cycle decomposition of G is a decomposition {G1, G2, . . . , Gt} where Gi is an mi-cycle for i = 1, 2, . . . , t. Obviously, necessary conditions for the existence of an (m1, m2, . . . , mt)-cycle decomposition of G are that 3 ≤ mi ≤ n for i = 1, 2, . . . , t and m1 +m2 + · · ·+mt = m. In [7] Bryant and Martin proved the following results for cycle decompositions of C n and C 3 n. Theorem 3.1 Let n ≥ 5 be an integer and let m1, m2, . . . , mt be a sequence of integers with mi ≥ 3 for i = 1, 2, . . . , t. Then C n = 〈{1, 2}〉n has an (m1, m2, . . . , mt)cycle decomposition if and only if each of the following conditions hold: (1) mi ≤ n for i = 1, 2, . . . , t; (2) m1 +m2 + · · ·+mt = 2n; and