Size, order, and connected domination

Abstract

We give a sharp upper bound on the size of a triangle-free graph of a given order and connected domination. Our bound, apart from strengthening an old classical theorem of Mantel and of Turán improves on a theorem of Sanchis. Further, as corollaries, we settle a long standing conjecture of Graffiti on the leaf number and local independence for triangle-free graphs and answer a question of Griggs, Kleitman, and Shastri on a lower bound of the leaf number in triangle-free graphs. © Canadian Mathematical Society 2013.