A Note on Polynomials and $f$-Factors of Graphs

Abstract

Let $G = (V,E)$ be a graph, and let $f : V \rightarrow 2^{\Bbb Z}$ be a function assigning to each $v \in V$ a set of integers in $\{0,1,2,\dots,d(v)\}$, where $d(v)$ denotes the degree of $v$ in $G$. Lovász defines an $f$-factor of $G$ to be a spanning subgraph $H$ of $G$ in which $d_{H}(v) \in f(v)$ for all $v \in V$. Using the combinatorial nullstellensatz of Alon, we prove that if $|f(v)| > \lceil {1\over 2}d(v) \rceil$ for all $v \in V$, then $G$ has an $f$-factor. This result is best possible and verifies a conjecture of Addario-Berry, Dalal, Reed and Thomason.

Highly edge‐connected factors using given lists on degrees

“…In this section, we first show the following theorem. Shirazi and Verstraëte [17] proved the case = 0 of Theorem 4, which was originally conjectured by Addario-Berry et al [1]. Note that their proof is based on Alon's Combinatorial Nullstellensatz [3], and hence it is not constructive.…”

“…In their proof, the following theorem played a central role. Note that this theorem appeared implicitly in [17] and the proof relying on the Combinatorial Nullstellensatz applies to this case with no changes.…”

On the Existence of General Factors in Regular Graphs

“…By using Alon's combinatorial nullstellensatz [4], Shirazi and Verstraëte [10] established the following brief result for general H-factors, which was originally posed by Addario-Berry et al [1] as a conjecture. Theorem 1.9 (Shirazi-Verstraëte).…”

Combinatorial Nullstellensatz Modulo Prime Powers and the Parity Argument

“…In Section 7, we focus on degree-constrained subgraphs modulo prime powers, and we will prove an analogous theorem for Shirazi-Verstraëte theorem [7].…”

“…By an F -avoiding subgraph we mean a subgraph ∅ = E ′ ⊆ E such that for every v ∈ V the number of incident edges of E ′ is not in F (v). Shirazi and Verstraëte [7] proved the following theorem. We give a new proof using our techniques.…”