The vertices and polygonal edges of the planar Archimedean tiling \(3^6\) of the plane is called the triangular tiling graph (TTG). A subgraph \(G\) of TTG is linearly convex if, for every line \(L\) which contains an edge of TTG, the set \(L \cap G\) is a (possibly degenerated or empty) line segment. A T-graph is any nontrivial, finite, linearly convex, 2-connected subgraph of TTG. The authors prove that with only one exeption, any T-graph contains a Hamiltonian cycle.