Hamiltonian cycles in certain graphs

Katherine Heinrich, W. D. Wallis
1978 Journal of the Australian Mathematical Society  
It is observed that arrays which arise in the scheduling of tournaments exist if and only if there are Hamiltonian cycles in certain graphs. The graphs are generalizations of those which arise in the "Footballers of Croam" problem. It is proven that such Hamiltonian cycles exist in infinite classes of the graphs. of the problem in the case g = 1 exists if and only if G nm has a Hamiltonian cycle. The case n = 2k-1, m = k-1, g = 1 has been considered by several authors. Meredith and Lloyd (1972)
more » ... th and Lloyd (1972) called G 2i ._ lfc _ 1 "the graph 0 k " and conjectured that, for k^4, 0 fc is an edge-disjoint union of Hamiltonian cycles plus, perhaps, a 1-factor. It is easily verified that 0 2 has a Hamiltonian cycle and that 0 3 does not (0 3 is in fact the Petersen graph). The graphs 0 4 , 0 5 , 0 6 , 0 7 and 0 8 have all been shown to have Hamiltonian cycles (see Balaban, 1972; Meredith and Lloyd, 1973; Mather, 1976) . The problem appears to be very difficult. For the rest of the paper we restrict our attention to the "graphical" case g = 1, but consider general m and n with n 3s 2m +1. (If n «S 2m the array is impossible and, in fact, the graph is disconnected, unless m = 1.) Without loss of generality we may choose the n-set 5 to be {1,2,..., «}. The authors wish to thank Mr. Ross Dunstan, who found the proof of Lemma 4.3.1, and also wish to thank the referee for helpful comments.
doi:10.1017/s1446788700011563 fatcat:eaidrkyf7je23fteqcq3ee6q3e