1-Homogeneous Graphs with Cocktail Party, mu -Graphs by Jurisic A.

By Jurisic A.

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U s } and B = { u l , u 2 , . . , u,}. Suppose the degree of Gallai theorems 43 u, E B in ( X ) is k + A,, i = 1, 2, . . ,t. Note that each A, > 0. Due to the minimality of (X),it follows that B is independent. Let Y denote the set of edges obtained from X by deleting A, edges from each u, E B, i = 1, 2, . . , t. Then for each v, E A, deg u, in ( Y ) is at most k, and for each u, E B deg u, in (Y)is exactly k. Counting the degrees of each u E V(G) in (X)and (Y)we get I sk + C (k + A,) + tk + ks - r=l I r=l = 2k(t I, + s) = 2kp.

Among the many conjectures which intrigue combinatorists nowadays, the following conjecture of Lovasz is undoubtedly one that attracts much attention. I Conjecture 1. [Lovasz] Every connected vertex-transitive graph has a Hamilton path. As Cayley graphs form a special class of vertex-transitive graphs, the above conjecture naturally leads to the following more specific conjecture: Conjecture 2. Every Cayley graph has a Hamilton cycle. V. C. Chen Though much effort has been made in order to verify the conjecture, yet up to date, it is known that the conjecture is only true for Cayley graphs over some very special classes of groups.

Once again, the only serious competitor here is the Kruskal-Katona bound [24]. To effect a comparison, we assume equal edge operation probabilities; the Kruskal-Katona bound requires this. Rather than content ourselves with small contrived examples, we compare the two bounds on the 1979 Arpanet, a real computer network whose analysis is of practical interest [3]; see Fig. 2. This network has 59 vertices and 71 edges. We should remark in advance that the s, t-cutsets produced by simple breadth-first search are not necessarily minimal; we always mod@ the edge-packing produced to make the cutsets minimal.

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