Optimal conditions for $(L_1;L_2)$ to be forcibly bigraphic

TL;DR

This paper establishes optimal conditions under which sequences of interval-based degree sequences are forcibly bigraphic, extending Gale and Ryser's classical theorem to interval sequences.

Contribution

It provides the first set of optimal necessary and sufficient conditions for interval sequences to be forcibly bigraphic, expanding the classical theory.

Findings

Derived two optimal conditions for interval sequences to be bigraphic.

Characterized when interval sequences are forcibly bigraphic.

Extended Gale-Ryser theorem to interval sequences.

Abstract

Let $L_1=([a_1,b_1],\ldots,[a_m,b_m])$ and $L_2=([c_1,d_1],\ldots,[c_n,d_n]$) be two sequences of intervals consisting of nonnegative integers with $b_1\ge \cdots\ge b_m$ and $d_1\ge \cdots\ge d_n$. In this paper, we first give two optimal conditions for the sequences of intervals $L_1$ and $L_2$ such that each pair $(P;Q)$ with $P=(p_1,\ldots,p_m)$, $Q=(q_1,\ldots,q_n)$, $a_i\le p_i\le b_i$ for $1\le i\le m$, $c_i\le q_i\le d_i$ for $1\le i\le n$ and $\sum\limits_{i=1}^m p_i=\sum\limits_{i=1}^n q_i$ is bigraphic. One of them is optimal sufficient condition and the other one optimal necessary condition. We also present a characterization of $(L_1;L_2)$ that is forcibly bigraphic on sequences of intervals. This is an extension of the well-known theorem on bigraphic sequences due to Gale and Ryser