A graph G is called interval colorable if it has a proper edge coloring with colors 1,2,3,… such that the colors of the edges incident to every vertex of G form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph G, denoted θint(G), which is the minimum number of interval colorable edge-disjoint subgraphs of G whose union is G. Our investigation is motivated by scheduling problems with compactness requirements, in particular, problems whose solution may consist of several schedules, but where each schedule must not contain any waiting periods or idle times for all involved parties. We first prove that every connected properly 3-edge colorable graph with maximum degree 3 is interval colorable, and using this result, we deduce an upper bound on θint(G) for general graphs G. We demonstrate that this upper bound can be improved in the case when G is bipartite, planar or complete multipartite and consider some applications in timetabling.

Let G(k) denote the set of connected k-regular graphs G, k≥2, where the number of vertices at distance 2 from any vertex in G does not exceed k. Asratian (2006) showed (using other terminology) that a graph G∈G(k) is Hamiltonian if for each vertex u of G the subgraph induced by the set of vertices at distance at most 2 from u is 2-connected. We prove here that in fact all graphs in the sets G(3), G(4) and G(5) are Hamiltonian. We also prove that the problem of determining whether there exists a Hamilton cycle in a graph from G(6) is NP-complete. Nevertheless we show that every locally connected graph G∈G(k), k≥6, is Hamiltonian and that for each non-Hamiltonian cycle C in G there exists a cycle C′ of length |V(C)|+ℓ in G, ℓ∈{1,2}, such that V(C)⊂V(C′). Finally, we note that all our conditions for Hamiltonicity apply to infinitely many graphs with large diameters.