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A Set Of Tournaments With Many Hamiltonian Cycles, Hayato Ushijima-Mwesigwa
A Set Of Tournaments With Many Hamiltonian Cycles, Hayato Ushijima-Mwesigwa
All Theses
For a random tournament on $3^n$ vertices, the expected number of Hamiltonian cycles is known to be $(3^n -1)!/2^{3^n}$. Let $T_1$ denote a tournament of three vertices $ {v_1, v_2, v_3}$. Let the orientation be such that there are directed edges from $v_1 $to $v_2$ , from $v_2$ to $v_3$ and from $v_3$ to $ v_1$. Construct a tournament $T_i$ by making three copies of $T_{i-1}$, $T_{i-1}'$, $T_{i-1}''$ and $T_{i-1}'''$. Let each vertex in $T_{i-1}'$ have directed edges to all vertices in $T_{i-1}''$, similarly place directed edges from each vertex in $T_{i-1}''$ to all vertices in $T_{i-1}'''$ and from $T_{i-1}'''$ …
Lefschetz Properties And Enumerations, David Cook Ii
Lefschetz Properties And Enumerations, David Cook Ii
Theses and Dissertations--Mathematics
An artinian standard graded algebra has the weak Lefschetz property if the multiplication by a general linear form induces maps of maximal rank between consecutive degree components. It has the strong Lefschetz property if the multiplication by powers of a general linear form also induce maps of maximal rank between the appropriate degree components. These properties are mainly studied for the constraints they place, when present, on the Hilbert series of the algebra. While the majority of research on the Lefschetz properties has focused on characteristic zero, we primarily consider the presence of the properties in positive characteristic. We study …