J. Comp. & Math. Sci. Vol.2 (6), 842-845 (2011)
Bound on the Minimum Cycle Cover Number of a Graph ALBERT WILLIAM and A. SHANTHAKUMARI Department of Mathematics, Loyola College, Chennai, India. ABSTRACT In this paper, we prove the existence of an edge-disjoint cycle cover for any graph G. The lower bound for the minimum edgedisjoint cycle cover number of a graph G is obtained. Keywords: Cycle cover number, cycle partition, circulant network.
1. INTRODUCTION The study of interconnection networks and their combinatorial properties has been pursued in the last decades with many different goals in mind and more recently has been encouraged by its applicability to large-scale parallel computer systems mainly because of their growing use in VLSI technologies3. The ring and chordal ring networks have many attractive properties like simplicity of structure, incremental extensibility, low valency, ease of implementation but highly vulnerable to faults in the network. Thus double-loop network or simply loop network structures came into existence. Loop networks are special cases of an important class of graphs called circulants. Cycles in interconnection networks are useful in many applications such as indexing, embedding linear arrays and rings, computing FFT and so on10. The problem of edge-disjoint cycle covers in a graph has been thoroughly studied both in the classical
complexity and in the approximation fields. It has a wide range of applications in computational biology4. The cycle partition of star graphs and arrangement graphs are studied in7,13. The uniform cycle partition problem and pancycle problem of butterfly networks are investigated in10. Circulant matrices have been employed for designing binary codes. Theoretical properties of circulant graphs have been studied extensively and surveyed in5. Every circulant graph is a vertex transitive graph and a Caley graph14. Most of the research concentrated on using the circulant graphs to build interconnection networks for distributed and parallel systems5. The network parameters to characterize the merits of interconnection schemes for multicomputer systems include: number of edges, values of diameter and average distances, symmetry, edge and node connectivities and extensibility. A Hamilton decomposition is a partition of the edge set into k Hamilton cycles if the graph is 2k-regular or k
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
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Albert William, et al., J. Comp. & Math. Sci. Vol.2 (6), 842-845 (2011)
Hamilton cycles and a perfect matching if the graph is (2k+1)-regular11. Alspach1 conjectured that every connected 2k-regular Cayley graph on a finite abelian group admits a Hamilton decomposition. For k = 1, the conjecture is trivially true, and Bermond et al.2 resolved the conjecture when k = 2.
called the jump sequence and the each sj, 1 ≤ j ≤ ⌊n/2⌋ is called a jump6. In this paper we obtain a lower bound for minimum edge-disjoint cycle cover number of a graph G. We also prove that the bound is sharp.
2. BASIC CONCEPTS Let V(G) and E(G) denote the vertex set and edge set of a graph G. Let H be a subgraph of G. Two subgraphs of G are said to be vertex-disjoint if they have no vertex in common and edge-disjoint if they have no edge in common. Given G(V, E), an edgedisjoint cycle cover is a partition of the edge set E(G) into a collection E₁, E₂, ..., Ek of pair wise disjoint subsets of E such that each Ei, 1 ≤ i ≤ k induces a cycle12. The minimum edge-disjoint cycle cover problem is to minimise k8. We denote this minimum number by η′(G). Minimum edge-disjoint cycle cover problem is NP-complete8. The degree dG(v) of a vertex v ∈ G is the number of edges of G incident with v, each loop counted as two edges9. A circulant undirected graph, denoted by G(n; ±{1, 2, ..., j}) where 1 ≤ j ≤ ⌊n/2⌋, n ≥ 3 is defined as an undirected graph consisting of the vertex set V ={0, 1, ..., n-1} and the edge set E = {(i, j) : ∣ j - i ∣ ≡ s(mod n), s ∈ {1, 2, ..., j}} [14]. The circulant graph shown in Figure 1 is G(8; ±{1,2}). It is clear that G(n; ±{1}) is the cycle Cn and G(n; ± {1, 2, ..., ⌊n/2⌋}) is the complete graph Kn. Clearly Cn = G(n; ±1) is a subgraph of G(n; ±{1, 2,..., ⌊n/2⌋}), G(n; ±S) is 2∣S∣- regular and ∣ E(G(n; ±{1, 2}) ∣ = 2n14. The sequence S = {1, 2, ..., ⌊n/2⌋} in the circulant network G(n; ±S), n ≥ 3 is
Figure 1: Edge-disjoint cycles of G(8;{1,2})
3. MAIN RESULTS Theorem 1 If G has an edge-disjoint cycle cover then the degree of every vertex of G is even. Proof. Suppose there exists a vertex v ∈ G of odd degree then it is not possible to cover all the edges incident at v with cycles. Theorem 2 Let G be a graph with an edgedisjoint cycle cover. Let ∆ be the maximum degree in G. Then η′(G) ≥ ∆ / 2. Proof. A cycle passing through v ∈ G contains exactly 2 edges of G incident at v. To cover all the edges incident at v, dG(v) / 2 cycles are required. Thus the minimum edge-disjoint cycle cover number η′ (G) ≥ ∆ / 2.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)
Albert William, et al., J. Comp. & Math. Sci. Vol.2 (6), 842-845 (2011)
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induces a cycle of length n, η′(G(n; ±{1, 2})) = 2. See Figure 2. 4. CONCLUSION
Figure 2: Edge-disjoint cycles of G(11;{1,2})
Theorem 2 shows that for circulant networks the minimum edge-disjoint cycle cover number η′(G(n; ±{1, 2, ..., j})) ≥ j. In particular, since a circulant network is a 2j-regular graph, we have η′(G(n; ±{1, 2, ..., j})) = j 11. Infact if n is relatively prime to one of the jumps then the edges corresponding to this jump form a Hamilton cycle. Algorithmically, the procedure in the following theorem shows that there are 2 Hamilton cycles that covers G(n; ±{1, 2}), n ≥ 5. Theorem 3. Let G(n; ±{1, 2}), n ≥ 5 be a circulant network and vertex labels be taken modulo n. Then η′(G(n; ±{1, 2})) = 2. Proof. Label the nodes of G(n; ±{1}) as 0, 1, 2, ..., n-1 in the clockwise sense. Let E = E₁ ∪ E₂ ... ∪ Ek be an edge-disjoint cycle cover of G(n; ±{1, 2}), n ≥ 5. Clearly η’(G(n; ±{1, 2})) ≠ 1. Let C(1) = (0, 1, 2..., n-3, n-1, n-2, 0) and C(2) = (0, n-1, 1, 3, ..., n-3, n-2, n-4, n-6, ..., 2, 0). E(C(1)) ∩ E(C(2)) = φ and ∣ E(C(1)) ∪ E(C(2)) ∣ = 2n = ∣ E(G(n; ±{1, 2})) ∣. Hence the cycles C(1) and C(2) are edge-disjoint. Since each C(i), 1 ≤ i ≤ 2
In this paper, a lower bound for minimum edge-disjoint cycle cover number of a graph has been derived. The bound is also proved to be sharp for Hamilton cycles. It would be interesting to obtain a graph G with minimum edge-disjoint cycle cover without using Hamilton cycles. REFERENCES 1. Alspach B., “Research Problem 59”, Discrete Mathematics, Vol. 59, 115 (1984). 2. Bermond J. C., Favaron O. and Maheo M., “Hamilton decomposition of Cayley graphs of degree four”, J. Combin. Theory Ser. B, Vol. 46 (1989). 3. Beivide R., Herrada E., Balcazar J. L. and Arruabarrena A., “Optimal distance Networks of Low Degree for Parallel Computers”, IEEE Transactions on Computers, Vol. 40 (1991). 4. Bafna V. and Pevzner P., “Genome Rearrangements and Sorting by Reversals”, SIAM J. Comput.,Vol. 25 (1996). 5. Bermond J. C., Comellas F. and Hsu D. F.,“Distributed loop computer networks, A Survey Journal of Parallel and Distributed Computing”, Vol. 24 (1995). 6. Boesch F. and Tindell R., “Circulants and Their Connectivities”, Journal of Graph Theory, Vol. 8, 487–499 (1984). 7. Day K. and Tripathi A., “Embedding of cycles in arrangement graphs”, IEEE Transactions on Computers, Vol. 12, 1002-1006 (1993).
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Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)