  We present eighteen exact analogs of six well-known fundamental Theorems (due
to Dirac, Nash-Williams and Jung) in hamiltonian graph theory providing
alternative compositions of graph invariants. In Theorems 1-3 we give three
lower bounds for the length of a longest cycle $C$ of a graph $G$ in terms of
minimum degree $\delta$, connectivity $\kappa$ and parameters $\bar{p}$,
$\bar{c}$ - the lengths of a longest path and longest cycle in $G\backslash C$,
respectively. These bounds have no analogs in the area involving $\bar{p}$ and
$\bar{c}$ as parameters. In Theorems 11 and 12 we give two Dirac-type results
for generalized cycles including a number of fundamental results (concerning
Hamilton and dominating cycles) as special cases. Connectivity invariant
$\kappa$ appears as a parameter in some fundamental results and in some their
exact analogs (Theorems 3-10) in the following chronological order: 1972
(Chv\'{a}tal and Erd\"{o}s), 1981a (Nikoghosyan), 1981b (Nikoghosyan), 1985a
(Nikoghosyan), 1985b (Nikoghosyan), 2000 (Nikoghosyan), 2005 (Lu, Liu, Tian),
2009 (Nikoghosyan), 2009a (Yamashita), 2009b (Yamashita), 2011a (Nikoghosyan),
2011b (Nikoghosyan).
