| Popis: |
If a cycle decomposition of a graph G admits two resolutions, and , such that for each resolution class and , then the resolutions and are said to be orthogonal. In this paper, we introduce the notion of an orthogonally resolvable cycle decomposition, which is a cycle decomposition admitting a pair of orthogonal resolutions. An orthogonally resolvable cycle decomposition of a graph G may be represented by a square array in which each cell is either empty or filled with a k–cycle from G, such that every vertex appears exactly once in each row and column of the array and every edge of G appears in exactly one cycle. We focus mainly on orthogonal k-cycle decompositions of and (the complete graph with the edges of a 1-factor removed), denoted . We give general constructions for such decompositions, which we use to construct several infinite families. We find necessary and sufficient conditions for the existence of an OCD(n, 4). In addition, we consider orthogonal cycle decompositions of the lexicographic product of a complete graph or cycle with . Finally, we give some nonexistence results. |
Plný text