IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARYFEBRUARY zyxwvuts 1993 237 0018-9472/93$03.00 zyxwvutsrq 0 1993 IEEE Determining Optimally Ordered Pairs Using Edge-Coloring of Graphs Ian Cloete and Wilma G. Cloete Abstmct-Tables to find optimally ordered pairs for the assessment of judgemental information were published. The optimality of such orders according to zyxwvutsrqponm spacing and balance requirements is proved using edge- colorings of graphs. In addition, the constructive proof leads to an elegant 1) Maintain the greatest possible spacing between any pairs involving the same altemative. For zyxw n odd the bounds for optimal spacing for all pairs is the following: Pairs involving the same alternative are separated by at most (n zyx - 1)/2 pairs, and by at least (n - 3)/2 pairs. Both the upper and the lower bound have to be attained for optimal spacing. 2) Present an alternative equally often as the first member and as the second member of a pair. Pairs are said to be balanced. For example, for zyxwvu n = 5 the 10 ordered pairs given by Ross are algorithm for constructing orders by hand without consulting tables. I. INTRODUCTION Determining relative preferences among different criteria (“stim- uli”) plays an important role in many areas of research. The assess- ment of judgemental information belongs traditionally to a branch of psychology known as psychometrics. The assumption is that relative preferences can be established if the criteria are lined up in pairs and presented for comparison. The order of pairs in the stimulus series is important since it influences judgement. In such an order the idea is to minimize judgemental bias by suitably arranging the order of comparisons because regular repetitions may influence judgement undesirably. In the order the greatest possible spacing should be achieved between pairs involving identical members, and the pairs should be balanced so that a given member of a pair does not occur more often on the left than on the right. One of the scientists who made an important contribution to the method of presenting pairs in an optimum order was Ross [l], [2]. This paper gives a detailed account that the Ross method for determining pairs of alternatives for comparison can be explained by edge-colorings of graphs. In search of a general algorithm, Ross gave a method that makes use of two tables to construct an order [l]. We present a simpler algorithm that chooses elements to compare based on their position on an imaginary circle. The advantage is that the order can easily be constructed by hand without consulting tables. The paper is organized as follows: Firstly, the Ross method is explained and an example is given. In Section 111 relevant terms of graph theory are given, and it is shown when and why perfect balance of pairs can be achieved. In Section IV the Ross method is developed in terms of edge-colorings of complete graphs. It is proved that the order is optimal with respect to the given criteria. Lastly, in Section V an algorithm for generating a Ross order for n elements is presented based ,on the properties obtained from graph theory. The algorithm is also suitable for computer implementa- tion. zyxwvutsrqpon 11. THE ROSS METHOD OF PAIRED COMPARISONS The method of paired comparisons, and in particular the Ross method, is explained as follows: Assume zyxwvutsrqpo n alternatives exist, each of which must be compared with every other alternative exactly once. This gives ( ) comparisons. The Ross method is concerned with the order in which these pairs should be presented for comparison. His order satisfies the following restrictions: Manuscript received December 22, 1990; revised April 20, 1992. The authors are with the Department of Computer Science, University of IEEE Log Number 9202128. Stellenbosch, Stellenbosch 7600, South Africa. He also suggests for n even that the order for n + 1 be determined first, and that all pairs involving the additional alternative then be eliminated. In the following therefore, consider only odd n, n 2 5. 111. DEFINITIONS AND GRAPH REPRESENTATION Before Ross’ method is developed by edge-colorings of graphs a few definitions [3], [4] are in order. Complete graphs on n vertices, denoted by zyxwvu li,, are simple graphs in which each pair of distinct vertices is joined by an edge. Edges are adjacent if they have a common vertex. A subset E, of the edge-set E of a graph G is called a matching if no two of its elements are adjacent. A matching E, of G is called a maximum matching if G has no matching E3 with IEjI > IEtl. An m-edge coloring of G is constructed by assigning 7n colors 1,2,. . . , m to the edges of G. This coloring is proper if no two adjacent edges have the same color. An m- edge coloring is also a partition (El, E2, . . . , E,) of E where E, is the set of edges assigned color i. Each E, is a matching if and only if they represent a proper m-edge coloring. The edge chromatic number zyxwvut X‘( G) is the minimum m for which G has a proper m-edge coloring. Pairs can be represented by a graph in which the vertices represent the alternatives and an edge represents the comparison between two alternatives. Because every alternative, say 2, is compared exactly once with each other alternative, say y, the comparison (I, y) can be represented as edges of a complete graph Kn with n alternatives or vertices. If an order satisfying restriction 1) has been constructed, the balance restriction 2) can be met in the following way: Because all the vertices of I<, (n odd) are of even degree IC, is Eulerian [3], i.e., I<, contains an Euler tour 00e1v1e2 ..‘emuO. Using this tour perfect balance can always be achieved. Iv. CONSTRUCHON OF AN ORDER In this section we determine the upper and lower bounds for optimal spacing and show how such an order is to be constructed. Using graph theoretic reasoning properties of the order to satisfy the spacing requirement are derived. The graph representation now allows the construction of an order satisfying two restrictions: a) An edge appears exactly once. b) No two consecutive edges in the order are adjacent. Restriction a) requires that alternatives be compared once only, while b) requires spacing between alternatives, i.e., vertices are not repeated in any two consecutive edges. The following argument derives the upper and lower bounds for maximum spacing.