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  • Cycle decompositions of ▫$K_n$▫ and ▫$K_n - I$▫ : thesis submitted in partial fulfillment of requirements for the degree of Doctor of Philosophy in the Department of Mathematics and Statistics
    Šajna, Mateja
    When does a complete graph admit a decomposition into cysles of some fixed length? Since the existence of such a decomposition requires that the degrees of all vertices be even, the complete graph ... must have an odd number of vertices. However, this questions can be extended to graphs with an even number of vertices in which every vertex has an even degree. A natural wey of creating such graphs that are very "close" to complete graphs is to remove a 1-factor from a complete graph with an even number of vertices. The question now becomes the following: when does ▫$K_n$▫ or ▫$K_n - I$▫, whichever is appropriate, admit a decomposition into cycles of a fixed length ▫$m$▫? There are two obvious necessary conditions, namely, that ▫$3 \le m \le n$▫ and that the cycle length ▫$m$▫ divides the number of edges in either ▫$K_n$▫, that is, ▫$\frac{n(n-1)}{2}$▫, or ▫$K_n - I$▫, that is, ▫$\frac{n(n-2)}{2}$▫. B. Alspach and H. Gavlas have shown that for the case when ▫$m$▫ and ▫$n$▫ are either both odd or both even, the necessary conditions are also sufficient. In this thesis we extend their results to the case ▫$m$▫ even, ▫$n$▫ odd, and ▫$m$▫ odd, ▫$n$▫ even. That is, we give a constructive proof of the following two statements: (1) ▫$K_n - I$▫ can be decomposed into cycles of length ▫$m$▫ whenever ▫$n$▫ is even, ▫$m$▫ is odd, ▫$3 \le m \le n$▫, and ▫$m$▫ divides ▫$\frac{n(n-2)}{2}$▫; and (2) ▫$K_n$▫ can be decomposed into cycles of length ▫$m$▫ whenever ▫$n$▫ is odd, ▫$m$▫ is even, ▫$3 \le m \le n$▫, and ▫$m$▫ divides ▫$\frac{n(n-1)}{2}$▫.
    Vrsta gradiva - disertacija
    Založništvo in izdelava - Vancouver : [M. Šajna], 1999
    Jezik - angleški
    COBISS.SI-ID - 9518169

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FMF in IMFM, Matematična knjižnica, Ljubljana Skladišče-Jadranska 19

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