Total domination and matching numbers in graphs with all vertices in triangles

A set MM of edges of a graph GG is a matching if no two edges in MM are incident to the same vertex. The matching number of GG is the maximum cardinality of a matching of GG. A set SS of vertices in GG is a total dominating set if every vertex of GG is adjacent to some vertex in SS. The minimum cardinality of a total dominating set of GG is the total domination number of GG. We prove that if all vertices of GG belong to a triangle, then the total domination number of GG is bounded above by its matching number. We in fact prove a slightly stronger result and as a consequence of this stronger result, we prove a Graffiti conjecture that relates the total domination and matching numbers in a graph.