Related papers: Nowhere-Zero $\vec k$-Flows on Graphs

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…

We initiate the study of a multivariate graph polynomial $\Phi_G(x,y,z)$ that interpolates between classical counting polynomials for matchings and for cycle structures arising in the Harary--Sachs expansion of the characteristic…

In this paper we propose a spectral flow for graph Laplacians, and prove that it counts the number of nodal domains for a given Laplace eigenvector. This extends work done for Laplacians on $\mathbb{R}^n$ to the graph setting. We mention…

We define a notion of equivariant non-degeneracy of $G$-maps to introduce the class of equivariantly non-degenerate flows on smooth compact manifolds with compact Lie group action. We prove genericity of this class and use this result to…

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…

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 $\phi_{d,p}(G)$…

The number of homomorphisms from a finite graph $F$ to the complete graph $K_n$ is the evaluation of the chromatic polynomial of $F$ at $n$. Suitably scaled, this is the Tutte polynomial evaluation $T(F;1-n,0)$ and an invariant of the cycle…

The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh that…

Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some…

Let G=(V,E) be a simple undirected graph. For a given set L of the real line, a function omega from E to L is called an L-flow. Given a vector gamma whose coordinates are indexed by V, we say that omega is a gamma-L-flow if for each v in V,…

Let X be a vector field on a compact connected manifold M. An important question in dynamical systems is to know when a function g:M -> R is a coboundary for the flow generated by X, i.e. when there exists a function f: M->R such that Xf=g.…

We consider the $Q_k$ flow on complete non-compact graphs. We prove that a complete graph evolves by the $Q_k$ curvature up to some time $T$ depending on the radius of a sphere enclosed by the initial graph.

Let $G=(V,E)$ be a graph with four distinguished vertices, two sources $s_1, s_2$ and two sinks $t_1,t_2$, let $c:\, E \rightarrow \mathbb Z_+$ be a capacity function, and let ${\cal P}$ be the set of all simple paths in $G$ from $s_1$ to…

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 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. We verify this conjecture for the class of flow-admissible signed graphs possessing a spanning even Eulerian subgraph, which includes as a…

A 1983 conjecture of Bouchet states that every flow-admissible signed graph has a nowhere-zero six-flow. We prove this conjecture for cyclically five-edge-connected, cubic signed graphs.

In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero $6$-flow. Bouchet himself proved that such signed graphs admit nowhere-zero $216$-flows and Zyka further proved that such signed graphs…

The one-way measurement model is a framework for universal quantum computation, in which algorithms are partially described by a graph G of entanglement relations on a collection of qubits. A sufficient condition for an algorithm to perform…

Let $G$ be a finite group and $\chi: G \rightarrow \mathbb{C}$ a class function. Let $H = (V,E)$ be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection $F$ of faces of $H$.…

We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximise the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from…