Nowhere‐Zero Flows on Signed Complete and Complete Bipartite Graphs

Bouchet's conjecture asserts that each signed graph which admits a nowhere‐zero flow has a nowhere‐zero 6‐flow. We verify this conjecture for two basic classes of signed graphs—signed complete and signed complete bipartite graphs by proving that each such flow‐admissible graph admits a nowhere‐zero 4‐flow and we characterise those which have a nowhere‐zero 2‐flow and a nowhere‐zero 3‐flow.     

Nowhere‐zero 3‐flows in products of graphs

A graph G is an odd‐circuit tree if every block of G is an odd length circuit. It is proved in this paper that the product of every pair of graphs G and H admits a nowhere‐zero 3‐flow unless G is an odd‐circuit tree and H has a bridge. This theorem is a partial result to the Tutte's 3‐flow conjecture and generalizes a result by Imrich and Skrekovski [7] that the product of two bipartite graphs admits a nowhere‐zero 3‐flow. A byproduct of this theorem is that every bridgeless Cayley graph G = Cay(Γ,S) on an abelian group Γ with a minimal generating set S admits a nowhere‐zero 3‐flow except for odd prisms. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Nowhere zero 4‐flow in regular matroids

Jensen and Toft 8 conjectured that every 2‐edge‐connected graph without a K₅‐minor has a nowhere zero 4‐flow. Walton and Welsh 19 proved that if a coloopless regular matroid M does not have a minor in {M(K_3,3), M*(K₅)}, then M admits a nowhere zero 4‐flow. In this note, we prove that if a coloopless regular matroid M does not have a minor in {M(K₅), M*(K₅)}, then M admits a nowhere zero 4‐flow. Our result implies the Jensen and Toft conjecture. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Nowhere‐zero flows in tensor product of graphs

In this paper, we characterize graphs whose tensor product admit nowhere‐zero 3‐flow. The main result is: For two graphs G₁ and G₂ with δ ≥ 2 and G₂ not belonging to a well‐characterized class of graphs, the tensor product of G₁ and G₂ admits a nowhere‐zero 3‐flow. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 284–292, 2007     

Nowhere‐zero 3‐flows in locally connected graphs

Let G be a graph. For each vertex v ∈V(G), N_v denotes the subgraph induces by the vertices adjacent to v in G. The graph G is locally k‐edge‐connected if for each vertex v ∈V(G), N_v is k‐edge‐connected. In this paper we study the existence of nowhere‐zero 3‐flows in locally k‐edge‐connected graphs. In particular, we show that every 2‐edge‐connected, locally 3‐edge‐connected graph admits a nowhere‐zero 3‐flow. This result is best possible in the sense that there exists an infinite family of 2‐edge‐connected, locally 2‐edge‐connected graphs each of which does not have a 3‐NZF. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211–219, 2003     

Nowhere‐Zero 5‐Flows On Cubic Graphs with Oddness 4

Tutte's 5‐flow conjecture from 1954 states that every bridgeless graph has a nowhere‐zero 5‐flow. It suffices to prove the conjecture for cyclically 6‐edge‐connected cubic graphs. We prove that every cyclically 6‐edge‐connected cubic graph with oddness at most 4 has a nowhere‐zero 5‐flow. This implies that every minimum counterexample to the 5‐flow conjecture has oddness at least 6.     

Bernstein-type Theorems in a Riemannian Manifold with an Irrotational Killing Vector Field

Mediterranean Journal of Mathematics - The minimal hypersurface equation for a graph in a Riemannian manifold which admits a nowhere zero Killing vector field, whose orthogonal distribution is...     

Zero‐sum flows in designs

Let D be a t ‐ ( v, k , λ) design and let N _i (D) , for 1 ≤ i ≤ t , be the higher incidence matrix of D , a ( 0 , 1 )‐matrix of size , where b is the number of blocks of D . A zero‐sum flow of D is a nowhere‐zero real vector in the null space of N ₁ ( D ). A zero‐sum k‐flow of D is a zero‐sum flow with values in { 1 , …, ±( k ? 1 )}. In this article, we show that every non‐symmetric design admits an integral zero‐sum flow, and consequently we conjecture that every non‐symmetric design admits a zero‐sum 5‐flow. Similarly, the definition of zero‐sum flow can be extended to N _i ( D ), 1 ≤ i ≤ t . Let be the complete design. We conjecture that N _t ( D ) admits a zero‐sum 3‐flow and prove this conjecture for t = 2 . © 2011 Wiley Periodicals, Inc. J Combin Designs 19:355‐364, 2011     

Circuit Covers of Signed Graphs

We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere‐zero integer flow. In the present article, we establish the existence of a universal coefficient such that every signed graph G that admits a nowhere‐zero integer flow has a signed circuit cover of total length at most . We show that if G is bridgeless, then , and in the general case .     

Circular flow on signed graphs

The circular flow number Φc(G,σ) of a signed graph (G,σ) is the minimum r for which an orientation of (G,σ) admits a circular r-flow. We prove that the circular flow number of a signed graph (G,σ) is equal to the minimum imbalance ratio of an orientation of (G,σ). We then use this result to prove that if G is 4-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) (as well as Φ(G,σ)) is at most 4. If G is 6-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) is strictly less than 4.     

Flows and Bisections in Cubic Graphs

A k‐weak bisection of a cubic graph G is a partition of the vertex‐set of G into two parts V₁ and V₂ of equal size, such that each connected component of the subgraph of G induced by () is a tree of at most vertices. This notion can be viewed as a relaxed version of nowhere‐zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere‐zero r‐flow has a ‐weak bisection. In this article, we study problems related to the existence of k‐weak bisections. We believe that every cubic graph that has a perfect matching, other than the Petersen graph, admits a 4‐weak bisection and we present a family of cubic graphs with no perfect matching that do not admit such a bisection. The main result of this article is that every cubic graph admits a 5‐weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5‐flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs that do contain bridges.     

Short cycle covers of graphs and nowhere‐zero flows

A shortest cycle cover of a graph G is a family of cycles which together cover all the edges of G and the sum of their lengths is minimum. In this article we present upper bounds to the length of shortest cycle covers, associated with the existence of two types of nowhere‐zero flows—circular flows and Fano flows. Fano flows, or Fano colorings, are nowhere‐zero ?‐flows on cubic graphs, with certain restrictions on the flow values meeting at a vertex. Such flows are conjectured to exist on every bridgless cubic graph. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 68:340‐348, 2011     

Circular flows of nearly Eulerian graphs and vertex‐splitting

The odd edge connectivity of a graph G, denoted by λ_o(G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and ? be a positive real number. We proved that there is a function f_S(?) (depends on the surface S and lim_?→0 f_S(?)=∞) such that any graph G embedded in S with the odd‐edge connectivity at least f_S(?) admits a nowhere‐zero circular (2+?)‐flow. Another major result of the work is a new vertex splitting lemma which maintains the old edge connectivity and graph embedding. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 147–161, 2002     

Initial boundary value problem for a class of non‐linear strongly damped wave equations

The paper studies the existence, asymptotic behaviour and stability of global solutions to the initial boundary value problem for a class of strongly damped non‐linear wave equations. By a H00.5ptk‐Galerkin approximation scheme, it proves that the above‐mentioned problem admits a unique classical solution depending continuously on initial data and decaying to zero as t→+∞as long as the non‐linear terms are sufficiently smooth; they, as well as their derivatives or partial derivatives, are of polynomial growth order and the initial energy is properly small. Copyright © 2003 John Wiley & Sons, Ltd.     

Nowhere zero flows in line graphs

Cai an Corneil (Discrete Math. 102 (1992) 103–106), proved that if a graph has a cycle double cover, then its line graph also has a cycle double cover, and that the validity of the cycle double cover conjecture on line graphs would imply the truth of the conjecture in general. In this note we investigate the conditions under which a graph G has a nowhere zero k-flow would imply that L(G), the line graph of G, also has a nowhere zero k-flow. The validity of Tutte's flow conjectures on line graphs would also imply the truth of these conjectures in general.     

Automorphism invariant Cartan subgroups and power maps of disconnected groups

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.     

Differentiable selection of optimal solutions in parametric linear programming

In the present paper we prove that if the data of a parametric linear optimization problem are smooth, the solution map admits a local smooth selection ``almost' everywhere. This in particular shows that the set of points where the marginal function of the problem is nondifferentiable is nowhere dense.

     

A Super Camassa–Holm Equation with N‐Peakon Solutions

A super Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero‐curvature equation, we derive a hierarchy of super Harry Dym type equations and establish their Hamiltonian structures. It is shown that the super Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N peakons. As an example, exact 1‐peakon solutions of the super Camassa–Holm equation are given. Infinitely many conserved quantities of the super Camassa–Holm equation and the super Harry Dym type equation are, respectively, obtained.     

Existence results for one‐dimensional fractional equations

In this note, a critical point result for differentiable functionals is exploited in order to prove that a suitable class of one‐dimensional fractional problems admits at least one non‐trivial solution under an asymptotical behaviour of the nonlinear datum at zero. A concrete example of an application is then presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Convergence of the interpolated coefficient finite element method for the two‐dimensional elliptic sine‐Gordon equations

An interpolated coefficient finite element method is presented and analyzed for the two‐dimensional elliptic sine‐Gordon equations with Dirichlet boundary conditions. It is proved that the discretization scheme admits at least one solution, and that a subsequence of the approximation solutions converges to an exact solution in L²‐norm as the mesh size tends to zero. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011