Conditions on subgraphs, degrees, and domination for hamiltonian properties of graphs

Motivated by a hamiltonicity result due to Ageev, involving claw-free graphs with a small domination number, we study analogous conditions that guarantee different types of hamiltonian properties. In our first main result we relax the condition on claw-freeness and obtain a common generalization of the result of Ageev and a recent result due to Cai. Our other results deal with similar conditions that guarantee traceability, pancyclicity and hamiltonian-connectivity. We also leave some open problems.