  We describe limits of line bundles on nodal curves in terms of toric
arrangements associated to Voronoi tilings of Euclidean spaces. These tilings
encode information on the relationship between the possibly infinitely many
limits, and ultimately give rise to a new definition of limit linear series.
This paper and its second and third companion parts are the first in a series
aimed to explore this new approach.
  In the present article, we set up the combinatorial framework and show how
graphs with integer lengths associated to the edges provide tilings of
Euclidean spaces by certain polytopes associated to the graph itself and to
certain of its subgraphs. We further provide a description of the combinatorial
structure of these polytopes and the way they are glued together in the tiling.
  In the second part of the series, we describe the arrangements of toric
varieties associated to these tilings. These results will be of use in the
third part to achieve our goal of describing all stable limits of a family of
line bundles along a degenerating family of curves.
