A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

Abstract

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. Grooming uniform all‐to‐all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G‐designs for five‐vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G‐designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five‐vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8‐groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9‐groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions.

Designs for graphs with six vertices and ten edges

“…Lemma 2.4. (i) There exist decompositions of complete multipartite graphs K 6,6,6,6,6 , K 4 6 , K 4 6 ,10 and K 15 9 into each of graphs n 1 , n 2 , n 4 , n 5 , n 7 , n 12 , n 14 and n 15 . (ii) There exist decompositions of K 5 5 ,10 and K 15 9 ,20 into each of graphs n 1 , n 2 , n 4 , n 5 , n 7 , n 12 and n 14 .…”

“…• Order 136 is constructed from decompositions of K 16 and K 15 9 , and the trivial 9-GDD of type 1 9 .…”

“…For details and references, see [2] and [5]. For more recent results, see [11], [17] and [9]. (ii) All graphs with six vertices and at most eight edges.…”

“…Appendix: DecompositionsProof of Lemma 2.1 K 16 With vertex set Z 15 ∪ {∞} the decompositions consist of (0, 2, 4, 8, 12, 10) 4 ,(6,2,10,14,18,4) 4 ,(8,16,12,6,10,18) 4 ,(14,16,18, 0,4,12) 4 ,(16,7,6,15,2,12) 5 ,(14, 0,7,17,10,19) 5 ,(5,14,13,11,2,12) 5 , With vertex set Z 25 the decompositions consist of (0, 7, 20, 21, 23, 16) 1 ,(14,20,5,9,3, 22) 1 ,(12,20,17,19,13,6) 1 ,(18,7,19,6, 22,13)…”

The spectrum problem for digraphs of order 4 and size 5

“…A G-decomposition of K n is also known as a (K n , G)-design. The set of all n for which K n admits a G-decomposition is called the spectrum of G. The spectrum has been determined for many classes of graphs, including all graphs on at most 4 vertices [3] and all graphs on 5 vertices (see [4] and [11]). We direct the reader to [2] and [5] for recent surveys on graph decompositions.…”