Hamiltonicity of <math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <msub> <mrow> <mn>3</mn> </mrow> <mrow> <mi>t</mi> </mrow> </msub> <mtext>EC</mtext> </math> Graphs with <math xmlns='http://www.w3.org/1998/Math/MathML' id='M2'> <mi>α</mi> <mo>=</mo> <mi>κ</mi> <mo>+</mo> <mn>1</mn> </math>

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S . The minimum cardinality of a total dominating set of G is the total domination number γ t G of G . The graph G is total domination edge-critical, or γ t EC , if for every edge e in the complement of G , γ t G + e < γ t G . If G is γ t EC and γ t G = k , we say that G is k t EC . In this paper, we show that every 3 t EC graph with δ G ≥ 2 and α G = κ G + 1 has a Hamilton cycle.