My Research Visiting Card in Hamiltonian Graph Theory

TL;DR

This paper introduces eighteen new theorems in Hamiltonian graph theory, providing alternative bounds and results involving graph invariants like minimum degree, connectivity, and parameters related to paths and cycles.

Contribution

It presents novel analogs of fundamental theorems, incorporating parameters like path and cycle lengths in relation to graph invariants, expanding the theoretical framework.

Findings

Eighteen exact analogs of classical theorems in Hamiltonian graph theory.

New bounds involving path and cycle lengths with no previous analogs.

Dirac-type results for generalized cycles including Hamilton and dominating cycles.

Abstract

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).