Related papers: Nowhere-Zero Flow Polynomials

Let $G$ be a graph. A zero-sum flow in $G$ is an assignment of nonzero real number to the edges such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be naturel number. A zero-sum $k$-flow is a flow with…

We generalize Tutte's integer flows and the $d$-dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajn\'{i}k, and Tabarelli to \emph{$d$-dimensional $p$-normed nowhere-zero flows} and define the corresponding flow index $\phi_{d,p}(G)$…

This paper concerns a generalization of nowhere-zero modular q-flows from graphs to simplicial complexes of dimension d greater than 1. A modular q-flow of a simplicial complex is an element of the kernel of the d-th boundary map with…

Lov\'{a}sz et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident…

In contrast to ordinary graphs, the number of the nowhere-zero group-flows in a signed graph may vary with different groups, even if the groups have the same order. In fact, for a signed graph $G$ and non-negative integer $d$, it was shown…

Bouchet conjectured in 1983 that every flow-admissible signed graph admits a nowhere-zero 6-flow which is equivalent to the restriction to cubic signed graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable cubic…

Let $\Gamma$ be a multigraph with for each vertex a cyclic order of the edges incident with it. For $n \geq 3$, let $D_{2n}$ be the dihedral group of order $2n$. Define $\mathbb{D} := \{(\begin{smallmatrix} 1 & a \\ 0 & 1 \end{smallmatrix})…

Let $G$ be a graph. A zero-sum flow of $G$ is an assignment of non-zero real numbers to the edges of $G$ such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be a natural number. A zero-sum $k$-flow is a…

We study the vanishing sets of slice regular polynomials in several quaternionic variables. We obtain a geometric description of the vanishing sets in two variables, which leads to a new version of the Strong Hilbert Nullstellensatz in the…

Let $Z_2\times Z_2=\{0, \alpha, \beta, \alpha+\beta\}$. If $G$ is a bridgeless cubic graph, $F$ is a perfect matching of $G$ and $\overline{F}$ is the complementary 2-factor of $F$, then a no-where zero $Z_2\times Z_2$-flow $\theta$ of…

Tutte's $3$-flow conjecture says that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. Kochol (2001) showed that it is enough to prove this conjecture for $5$-edge-connected graphs. Former, Jaeger, Linial, Payan, and Tarsi…

Given a $t$-$(v, k, \lambda)$ design, $\mathcal{D}=(X,\mathcal{B})$, a zero-sum $n$-flow of $\mathcal{D}$ is a map $f : \mathcal{B}\longrightarrow \{\pm1,\ldots, \pm(n-1)\}$ such that for any point $x\in X$, the sum of $f$ over all blocks…

We define a trivariate polynomial combining the NL-coflow and the NL-flow polynomial, which build a dual pair counting acyclic colorings of directed graphs, in the more general setting of regular oriented matroids.

This article studies real roots of the flow polynomial $F(G,\lambda)$ of a bridgeless graph $G$. For any integer $k\ge 0$, let $\xi_k$ be the supremum in $(1,2]$ such that $F(G,\lambda)$ has no real roots in $(1,\xi_k)$ for all graphs $G$…

Given a finite directed acyclic graph, the space of non-negative unit flows is a lattice polytope called the flow polytope of the graph. We consider the volumes of flow polytopes for directed acyclic graphs on $n+1$ vertices with a fixed…

Tutte conjectured in 1972 that every 4-edge connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge-connected graph has an edge orientation in which every out-degree…

A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a…

A set $R\subseteq E(G)$ of a graph $G$ is $k$-removable if $G-R$ has a nowhere-zero $k$-flow. We prove that every graph $G$ admitting a nowhere-zero $4$-flow has a $3$-removable subset consisting of at most $\frac{1}{6}|E(G)|$ edges. This…

By studying the weak closure of multidimensional off-diagonal self-joinings we provide a criterion for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid…

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic…