Coloring linear hypergraphs: the Erdős–Faber–Lovász conjecture and the Combinatorial Nullstellensatz

The long-standing Erdős–Faber–Lovász conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using n colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős–Faber–Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM

Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

Some new results about a conjecture by Brian Alspach

In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of $$\mathbb {Z}_n{\setminus } \{0\}$$ Z n \ { 0 } of size k such that $$\sum _{z\in A} z\not = 0$$ ∑ z ∈ A z ≠ 0 , it is possible to find an ordering $$(a_1,\ldots ,a_k)$$ ( a 1 , … , a k ) of the elements of A such that the partial sums $$s_i=\sum _{j=1}^i a_j$$ s i = ∑ j = 1 i a j , $$i=1,\ldots ,k$$ i = 1 , … , k , are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $$k\le 11$$ k ≤ 11 in cyclic groups of prime order. Here, we extend this result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $$\mathbb {Z}_n$$ Z n . We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of $$\mathbb {Z}_p{\setminus }\{0\}$$ Z p \ { 0 } , where p is a prime, there exists an ordering of the elements of A such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis, and Schmitt, based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets A of size 12.