Kirby diagrams and 5-colored graphs representing compact 4-manifolds

It is well-known that in dimension 4 any framed link (L, c) uniquely represents the PL 4-manifold M^4 (L, c) obtained from D^4 by adding 2-handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram (L^(*), d)), the associated PL 4-manifold M^4(L^(*), d) is obtained from D^4 by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored graphs: in particular, we describe how to construct algorithmically a (regular) 5-colored graph representing M^4(L^(*), d), directly "drawn over" a planar diagram of (L^(*), d), or equivalently how to algorithmically obtain a triangulation of M^4(L^(*),d). As a consequence, the procedure yields triangulations for any closed (simply-connected) PL 4-manifold admitting handle decompositions without 3-handles. Furthermore, upper bounds for both the invariants gem-complexity and regular genus of M^4(L^(*), d) are obtained, in terms of the combinatorial properties of the Kirby diagram.