Nowhere-zero 3-flows of highly connected graphs

Let G be a k-edge-connected graph of order n. If k4log₂ n then G has a nowhere-zero 3-flow.     

Nowhere-zero 6-flows

We prove that every graph with no isthmus has a nowhere-zero 6-flow, that is, a circulation in which the value of the flow through each edge is one of ±1, ±2,…, ±5. This improves Jaeger's 8-flow theorem, and approaches Tutte's 5-flow conjecture.     

Nowhere-zero 3-flows and modulo k-orientations

The main theorem of this paper provides partial results on some major open problems in graph theory, such as Tutte?s 3-flow conjecture (from the 1970s) that every 4-edge connected graph admits a nowhere-zero 3-flow, the conjecture of Jaeger, Linial, Payan and Tarsi (1992) that every 5-edge-connected graph is _Z3$Z_3$-connected, Jaeger?s circular flow conjecture (1984) that for every odd natural number k?3$k?3$, every (2k−2)$(2k-2)$-edge-connected graph has a modulo k  -orientation, etc. It was proved recently by Thomassen that, for every odd number k?3$k?3$, every (2k²+k)$(2k^2+k)$-edge-connected graph G has a modulo k-orientation; and every 8-edge-connected graph G   is _Z3$Z_3$-connected and admits therefore a nowhere-zero 3-flow. In the present paper, Thomassen?s method is refined to prove the following: For every odd number  k?3$k?3$, every  (3k−3)$(3k-3)$-edge-connected graph has a modulo k-orientation. As a special case of the main result, every 6-edge-connected graph is  _Z3$Z_3$-connected and admits therefore a nowhere-zero 3-flow. Note that it was proved by Kochol (2001) that it suffices to prove the 3-flow conjecture for 5-edge-connected graphs.     

Nowhere-zero 15-flow in 3-edge-connected bidirected graphs

It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero κ flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet＇s conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.     

Nowhere-zero flows in low genus graphs

A nowhere-zero k-flow is an assignment of edge directions and integer weights in the range 1,…, k ? 1 to the edges of an undirected graph such that at every vertex the flow in is equal to the flow out. Tutte has conjectured that every bridgeless graph has a nowhere-zero 5-flow. We show that a counterexample to this conjecture, minimal in the class of graphs embedded in a surface of fixed genus, has no face-boundary of length <7. Moreover, in order to prove or disprove Tutte's conjecture for graphs of fixed genus γ, one has to check graphs of order at most 28(γ ? 1) in the orientable case and 14(γ ? 2) in the nonorientable case. So, in particular, it follows immediately that every bridgeless graph of orientable genus ?1 or nonorientable genus ?2 has a nowhere-zero 5-flow. Using a computer, we checked that all graphs of orientable genus ?2 or nonorientable genus ?4 have a nowhere-zero 5-flow.     

Nowhere-zero eigenvectors of graphs

A vector is called nowhere-zero if it has no zero entry. In this article we search for graphs with nowhere-zero eigenvectors. We prove that distance-regular graphs and vertex-transitive graphs have nowhere-zero eigenvectors for all of their eigenvalues and edge-transitive graphs have nowhere-zero eigenvectors for all non-zero eigenvalues. Among other results, it is shown that a graph with three distinct eigenvalues has a nowhere-zero eigenvector for its smallest eigenvalue.     

Mixing 3-colourings in bipartite graphs

For a 3-colourable graph G, the 3-colour graph of G, denoted , is the graph with node set the proper vertex 3-colourings of G, and two nodes adjacent whenever the corresponding colourings differ on precisely one vertex of G. We consider the following question: given G, how easily can one decide whether or not is connected? We show that the 3-colour graph of a 3-chromatic graph is never connected, and characterise the bipartite graphs for which is connected. We also show that the problem of deciding the connectedness of the 3-colour graph of a bipartite graph is coNP-complete, but that restricted to planar bipartite graphs, the question is answerable in polynomial time.     

On zero-sum -flows of graphs

For a graph G, a zero-sum flow is an assignment of non-zero real numbers on the edges of G such that the total sum of all edges incident with any vertex of G is zero. A zero-sum k-flow for a graph G is a zero-sum flow with labels from the set {±1,…,±(k-1)}. In this paper for a graph G, a necessary and sufficient condition for the existence of zero-sum flow is given. We conjecture that if G is a graph with a zero-sum flow, then G has a zero-sum 6-flow. It is shown that the conjecture is true for 2-edge connected bipartite graphs, and every r-regular graph with r even, r>2, or r=3.     

Non-hamiltonian 3-connected cubic bipartite graphs

A non-hamiltonian cyclically 4-edge-connected bicubic graph with 54 vertices is constructed. This is the smallest non-hamiltonian 3-connected bicubic graph known, and is the first such graph that is cyclically 4-edge-connected.     

Nowhere-zero integral chains and flows in bidirected graphs

General results on nowhere-zero integral chain groups are proved and then specialized to the case of flows in bidirected graphs. For instance, it is proved that every 4-connected (resp. 3-connected and balanced triangle free) bidirected graph which has at least an unbalanced circuit and a nowhere-zero flow can be provided with a nowhere-zero integral flow with absolute values less than 18 (resp. 30). This improves, for these classes of graphs, Bouchet's 216-flow theorem (J. Combin. Theory Ser. B 34 (1982), 279–292). We also approach his 6-flow conjecture by proving it for a class of 3-connected graphs. Our method is inspired by Seymour's proof of the 6-flow theorem (J. Combin. Theory Ser. B 30 (1981), 130–136), and makes use of new connectedness properties of signed graphs.     

Nowhere-zero 4-flow in almost Petersen-minor free graphs

Tutte [W.T. Tutte, On the algebraic theory of graph colorings, J. Combin. Theory 1 (1966) 15-20] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let be the graph obtained from the Petersen graph by contracting μ edges from a perfect matching. In this paper we prove that every bridgeless -minor free graph admits a nowhere-zero 4-flow.     

On 3-colorings of bipartite p-threshold graphs

In this paper some extremal properties of 3-colorings of bipartite complete graphs in the class of all bipartite p-threshold graphs that are uniquely 2-colorable are proved. As a consequence it is shown that the complete bipartite graphs K_{p, p + r} where p ? 2 and 0 ? r < are chromatically unique. A useful result concerning the maximization of a sum of powers of two under certain restrictions, which has an arithmetical interest, is also presented.     

Cycles in bipartite graphs

For k an integer, let G(a, b, k) denote a simple bipartite graph with bipartition (A, B) where |A| = a ≥ 2, |B| = b ≥ k ≥ 2, and each vertex of A has degree at least k. We prove two results concerning the existence of cycles in G(a, b, k).     

Domination in bipartite graphs

We prove that the domination number of a graph of order n and minimum degree at least 2 that does not contain cycles of length 4, 5, 7, 10 or 13 is at most . Furthermore, we derive upper bounds on the domination number of bipartite graphs of given minimum degree.     

Nested bipartite graphs

We investigate a class of bipartite graphs, whose structure is determined by a binary number. The work for this research was supported by the Max Kade Foundation.