Cycle covers of graphs with a nowhere-zero 4-flow

It is shown that the edges of a simple graph with a nowhere-zero 4-flow can be covered with cycles such that the sum of the lengths of the cycles is at most |E(G)| + |V(G)| ?3. This solves a conjecture proposed by G. Fan.     

Smallest (1, 2)-eulerian weight and shortest cycle covering

The concept of a (1, 2)-eulerian weight was introduced and studied in several papers recently by Seymour, Alspach, Goddyn, and Zhang. In this paper, we proved that if G is a 2-connected simple graph of order n (n ≧ 7) and w is a smallest (1, 2)-eulerian weight of graph G, then |E_w=even | n - 4, except for a family of graphs. Consequently, if G admits a nowhere-zero 4-flow and is of order at least 7, except for a family of graphs, the total length of a shortest cycle covering is at most | V(G) | + |E(G) |- 4. This result generalizes some previous results due to Bermond, Jackson, Jaeger, and Zhang.     

The size of graphs without nowhere-zero 4-flows

Let G be a 2-edge-connected simple graph with order n. We show that if | V(G)| ≤ 17, then either G has a nowhere-zero 4-flow, or G is contractible to the Petersen graph. We also show that for n large, if | V(G)| n ? 17/2 + 34, then either G has a nonwhere-zero 4-flow, or G can be contracted to the Petersen graph. © 1995 John Wiley & Sons, Inc.     

Flows in 3-edge-connected bidirected graphs

It was conjectured by A. Bouchet that every bidirected graph which admits a nowhere-zero k-flow admits a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. O. Zyka improved the result with 6 replaced by 30. R. Xu and C. Q. Zhang showed that the conjecture is true for 6-edge-connected graph, which is further improved by A. Raspaud and X. Zhu for 4-edge-connected graphs. The main result of this paper improves Zyka’s theorem by showing the existence of a nowhere-zero 25-flow for all 3-edge-connected graphs.     

Decomposition of the Flow Polynomial

The flow polynomials denote the number of nowhere-zero flows on graphs, and are related to the well-known Tutte polynomials and chromatic polynomials. We will show the decomposition of the flow polynomials by edge-cuts and vertex-cuts of size 2 or 3. Moreover by using this decomposition, we will consider what kind of graphs have the same flow polynomials. Another application of the decomposition results is that if a bridgeless graph G does not admit a nowhere-zero k-flow and G has a small vertex- or edge-cut, then a proper bridgeless subgraph of G (a graph minor) does not admit a nowhere-zero k-flow either.     

Tutte's 5-flow conjecture for graphs of nonorientable genus 5

We develop four constructions for nowhere-zero 5-flows of 3-regular graphs that satisfy special structural conditions. Using these constructions we show a minimal counter-example to Tutte's 5-Flow Conjecture is of order ≥44 and therefore every bridgeless graph of nonorientable genus ≤5 has a nowhere-zero 5-flow. One of the structural properties is formulated in terms of the structure of the multigraph G(F) obtained from a given 3-regular graph G by contracting the cycles of a 2-factor F in G. © 1996 John Wiley & Sons, Inc.     

Tutte's 5-flow conjecture for the projective plane

Heawood proved that every planar graph with no 1-cycles is vertex 5-colorable, which is equivalent to the statement that every planar graph with no 1-bonds has a nowhere-zero 5-flow. Tutte has conjectured that every graph with no 1-bonds has a nowhere-zero 5-flow. We show that Tutte's 5-Flow Conjecture is true for all graphs embeddable in the real projective plane.     

On Group Connectivity of Graphs

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero Z ₃-flow and Jaeger et al. [Group connectivity of graphs–a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165-182] further conjectured that every 5-edge-connected graph is Z ₃-connected. These two conjectures are in general open and few results are known so far. A weaker version of Tutte’s conjecture states that every 4-edge-connected graph with each edge contained in a circuit of length at most 3 admits a nowhere-zero Z ₃-flow. Devos proposed a stronger version problem by asking if every such graph is Z ₃-connected. In this paper, we first answer this later question in negative and get an infinite family of such graphs which are not Z ₃-connected. Moreover, motivated by these graphs, we prove that every 6-edge-connected graph whose edge set is an edge disjoint union of circuits of length at most 3 is Z ₃-connected. It is a partial result to Jaeger’s Z ₃-connectivity conjecture. Received: May 23, 2006. Final version received: January 13, 2008     

Six-flows on almost balanced signed graphs

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. By Seymour's 6-flow theorem, Bouchet's conjecture holds for signed graphs with all edges positive. Recently, Rollová et al proved that every flow-admissible signed cubic graph with two negative edges admits a nowhere-zero 7-flow, and admits a nowhere-zero 6-flow if its underlying graph either contains a bridge, or is 3-edge-colorable, or is critical. In this paper, we improve and extend these results, and confirm Bouchet's conjecture for signed graphs with frustration number at most two, where the frustration number of a signed graph is the smallest number of vertices whose deletion leaves a balanced signed graph.     

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.     

Degree sum and nowhere-zero 3-flows

Let G be a simple graph on n vertices. In this paper, we prove that if G satisfies the condition that d(x)+d(y)≥n for each xy∈E(G), then G has no nowhere-zero 3-flow if and only if G is either one of the five graphs on at most 6 vertices or one of a very special class of graphs on at least 6 vertices.     

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.     

Tutte’s 5-flow conjecture for highly cyclically connected cubic graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let ω(G) be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph G. Tutte’s conjecture is equivalent to its restriction to cubic graphs with ω≥2. We show that if a cubic graph G has no edge cut with fewer than edges that separates two odd cycles of a minimum 2-factor of G, then G has a nowhere-zero 5-flow. This implies that if a cubic graph G is cyclically n-edge connected and , then G has a nowhere-zero 5-flow.     

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 5-Flows and Even (1,2)-Factors

A graph G = (V, E) admits a nowhere-zero k-flow if there exists an orientation H = (V, A) of G and an integer flow ${\varphi:A \to \mathbb{Z}}$ such that for all ${a \in A, 0 < |\varphi(a)| < k}$ . Tutte conjectured that every bridgeless graphs admits a nowhere-zero 5-flow. A (1,2)-factor of G is a set ${F \subseteq E}$ such that the degree of any vertex v in the subgraph induced by F is 1 or 2. Let us call an edge of G, F-balanced if either it belongs to F or both its ends have the same degree in F. Call a cycle of G F-even if it has an even number of F-balanced edges. A (1,2)-factor F of G is even if each cycle of G is F-even. The main result of the paper is that a cubic graph G admits a nowhere-zero 5-flow if and only if G has an even (1,2)-factor.     

The 7/5-conjecture strengthens itself

The following is proved: if every bridgeless graph G has a cycle cover of length at most 7/5|E(G)|, then every bridgeless graph G has a cycle cover of length at most 7/5|E(G)| such that any edge of G is covered once or twice. © 1995 John Wiley & Sons, Inc.     

Short cycle covers of cubic graphs

Let G be a bridgeless cubic graph. We prove that the edges of G can be covered by circuits whose total length is at most (44/27) |E(G)|, and if Tutte's 3-flow Conjecture is true, at most (92/57) |E(G)|.     

Flows, flow-pair covers and cycle double covers

In this paper, some earlier results by Fleischner [H. Fleischner, Bipartizing matchings and Sabidussi’s compatibility conjecture, Discrete Math. 244 (2002) 77-82] about edge-disjoint bipartizing matchings of a cubic graph with a dominating circuit are generalized for graphs without the assumption of the existence of a dominating circuit and 3-regularity. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow parity-pair-cover of G if the union of their supports covers the entire graph; f₁ is an h-flow and f₂ is a k-flow, and . Then G admits a nowhere-zero 6-flow if and only if G admits a (4,3)-flow parity-pair-cover; and G admits a nowhere-zero 5-flow if G admits a (3,3)-flow parity-pair-cover. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow even-disjoint-pair-cover of G if the union of their supports covers the entire graph, f₁ is an h-flow and f₂ is a k-flow, and for each {i,j}={1,2}. Then G has a 5-cycle double cover if G admits a (4,4)-flow even-disjoint-pair-cover; and G admits a (3,3)-flow parity-pair-cover if G has an orientable 5-cycle double cover.     

图的字典积的非零整数流(英文)

充分利用图的字典积的结构证明了以下结论:如果图G_1的每连通分支都非平凡,图G_2的阶数大于3,那么它们的字典积G_1[G_2]具有非零3-流.     

Join of two graphs admits a nowhere-zero 3-flow

Let G be a graph, and λ the smallest integer for which G has a nowherezero λ-flow, i.e., an integer λ for which G admits a nowhere-zero λ-flow, but it does not admit a (λ ? 1)-flow. We denote the minimum flow number of G by Λ(G). In this paper we show that if G and H are two arbitrary graphs and G has no isolated vertex, then Λ(G ∨ H) ? 3 except two cases: (i) One of the graphs G and H is K ₂ and the other is 1-regular. (ii) H = K ₁ and G is a graph with at least one isolated vertex or a component whose every block is an odd cycle. Among other results, we prove that for every two graphs G and H with at least 4 vertices, Λ(G ∨ H) ? 3.