Extremal Theorems for Degree Sequence Packing and the Two-Color Discrete Tomography Problem

A sequence π = (d1, . . . , dn) is graphic if there is a simple graph G with vertex set {v1, . . . , vn} such that the degree of vi is di. We say that graphic sequences π1 = (d (1) 1 , . . . , d (1) n ) and π2 = (d (2) 1 , . . . , d (2) n ) pack if there exist edge-disjoint n-vertex graphs G1 and G2 such that for j ∈ {1, 2}, dGj (vi) = d (j) i for all i ∈ {1, . . . , n}. Here, we prove several extremal degree sequence packing theorems that parallel central results and open problems from the graph packing literature. Specifically, the main result of this paper implies a degree sequence packing analogue to the widely studied BollobásEldridge-Catlin graph packing conjecture, along with a degree sequence version of the classical graph packing theorem of Sauer and Spencer. In discrete tomography, a branch of discrete imaging science, the goal is to reconstruct discrete objects using data acquired from low-dimensional projections. Specifically, in the k-color discrete tomography problem the goal is to color the entries of an m× n matrix using k colors so that each row and column receive a prescribed number of entries of each color. This problem is equivalent to packing the degree sequences of k bipartite graphs with parts of sizes m and n. Here we modify our techniques to prove several Sauer-Spencer-type theorems that have direct applications to the 2-color discrete tomography problem.