NOTICE: Due to a lapse in annual appropriations, most of this website is not being updated. Learn more.
Form submissions will still be accepted but will not receive responses at this time. Sections of this site for programs using non-appropriated funds (such as NVLAP) or those that are excepted from the shutdown (such as CHIPS and NVD) will continue to be updated.
An official website of the United States government
Here’s how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock (
) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.
New Results on Packing Degree Sequences and the 2-Color Discrete Tomography Problem
Published
Author(s)
James Shook, Michael Ferrara, Jennifer Diemunsch, Sogol Jahanbekam
Abstract
A sequence $\pi=(d_1,\ldots,d_n)$ is graphic if there is a simple graph $G$ with vertex set $\{v_1,\ldots,v_n\}$ such that the degree of $v_i$ is the $i$th entry of $\pi$. We say that graphic sequences $\pi_1=(d_1^{(1)},\ldots,d_n^{(1)})$ and $\pi_2=(d_1^{(2)},\ldots,d_n^{(2)})$, \emph{pack} if there exist edge-disjoint $n$-vertex graphs $G_1$ and $G_2$ such that for $j=1,2$, $d_{G_j}(v_i)=d_i^{(j)}$ for all $i=1,\ldots,n$. Here we give conditions on $\pi_1$ and $\pi_2$ to guarantee that these sequences pack. In particular, let $\Δ_j$ be the maximum degree of $\pi_j$, then if $\Δ_1\Δ_2<\frac{n}{2}$ then $\pi_1$ and $\pi_2$ pack. This result is a degree sequence analogue to a well-known result of Sauer and Spencer. Packing bigraphic sequences $\pi_1$ and $\pi_2$ applies to discrete tomography. In discrete tomography, the goal is to reconstruct discrete objects from low-dimensional projections. Specifically, we consider an $m\times n$ matrix, with the goal of coloring the entries with $k$ colors so that each row and column receive a prescribed numbers of entries of each colors. This problem is the same as packing degree sequences of $k$ bipartite graphs with parts of sizes $m$ and $n$. Here we show that two bigraphic sequences pack if $\Δ_1\Δ_2<\frac{n+m}{4}$.
Shook, J.
, Ferrara, M.
, Diemunsch, J.
and Jahanbekam, S.
(2015),
New Results on Packing Degree Sequences and the 2-Color Discrete Tomography Problem, Siam Journal on Discrete Mathematics, [online], https://doi.org/10.1137/140987912
(Accessed October 6, 2025)