An important property of graph which concerns their applications to networks is Hamiltonicity. Determining if a graph is hamiltonian is a NP-complete problem and no satisfactory characterization exists. Nevertheless, many sufficient conditions were found, usually expressed in terms of degree, connectivity, density, toughness, independent set, regularity and forbidden subgraphs. In 1973, Goodman and Hedetniemi gave two alternative sufficient conditions uniquely based on the possibility to decompose in some ways the graph into cliques. We first generalise one of these two conditions. We give a new clique decomposition condition which proves the hamiltonicity of a broader class of graphs. Then, we present a polynomial algorithm which decides the existence of such a decomposition for simplicial-connected graphs. A graph is simplicial-connected if every vertex is connected by a walk to a simplicial one. In particular, each connected graph containing a simplicial vertex is simplicial-connected, and so is every chordal graph