Abstract:Squashed 6-cycle systems are introduced as a natural counterpart to 2-perfect 6-cycle systems. The spectrum for the latter has been determined previously in [5]. We determine completely the spectrum for squashed 6-cycle systems, and also for squashed 6-cycle packings.

The purpose of this paper is the initiation of an attack on the general existence problem for almost resolvable 2k-cycle systems. We give a complete solution for 2k 5 6 as well as a complete solution modulo one possible exception for 2k 5 10 and 14. We also show that the existence question for almost resolvable 2k-cycle systems can be settled if we can show the existence for the two smallest possible orders 4k11 and 8k11.

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A proper edge-coloring of a graph defines at each vertex the set of colors of its incident edges. This set is called the palette of the vertex. In this paper we are interested in the minimum number of palettes taken over all possible proper colorings of a graph.