Connected domination number and traceable graphs

Abstract

Let G be a simple connected graph with minimum degree δ , second minimum degree δ (cid:48) , and connected domination number γ c ( G ) . It is shown that G has a spanning path whenever γ c ( G ) ≥ n − δ (cid:48) − 1 . This result is best possible for δ (cid:48) < 3 ; that is, if γ c ( G ) ≥ n − δ (cid:48) − 2 and δ (cid:48) < 3 , then G may or may not contain a spanning path. Also, this result settles completely a conjecture posed recently by Chellali and Favaron. In addition, for every choice of δ (cid:48) and δ , an infinite family of non-traceable graphs satisfying δ (cid:48) > δ and γ c ( G ) ≤ n − 2 δ (cid:48) is provided, which shows that if another recent conjecture by Chellali and Favaron is true, then it is best possible in a sense. The obtained results, apart from addressing some stronger versions of conjectures generated by the computer program Graffiti.pc, improve some known results.