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.     

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.     

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 3-flow of graphs with small independence number

Tutte’s $3$-flow conjecture states that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In this paper, we characterize all graphs with independence number at most $4$ that admit a nowhere-zero $3$-flow. The characterization of $3$-flow verifies Tutte’s $3$-flow conjecture for graphs with independence number at most $4$ and with order at least $21$. In addition, we prove that every odd-$5$-edge-connected graph with independence number at most $3$ admits a nowhere-zero $3$-flow. To obtain these results, we introduce a new reduction method to handle odd wheels.     

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 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.     

Group connectivity in line graphs

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A|≥k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b:V(G)?A with ∑_v∈V(G)b(v)=0, there always exists a map f:E(G)?A−{0}, such that at each v∈V(G), in A, then G is A-connected. Let Z₃ denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z₃-connected. In this paper, we proved the following.

• (i) 

  Every 5-edge-connected graph is Z3-connected if and only if every 5-edge-connected line graph is Z3-connected.

• (ii) 

  Every 6-edge-connected triangular line graph is Z3-connected.

• (iii) 

  Every 7-edge-connected triangular claw-free graph is Z3-connected.

In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere zero 3-flow.     

Almost all 5-regular graphs have a 3-flow

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 in-degree is either 1 or 4. We show that the assertion of the conjecture holds asymptotically almost surely for random 5-regular graphs. It follows that the conjecture holds for almost all 4-edge–connected 5-regular graphs.     

Nowhere-zero 3-flows in Cayley graphs on generalized dihedral group and generalized quaternion group

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111: 416–419]. We also generalizes an early result of M. Nánásiová and M. ?koviera [J. Algebraic Combin., 2009, 30: 103–110].     

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.     

NASH EQUILIBRIUM OF TWO GAME MODELS ON THE RANDOM SOCIAL NETWORK

A graph $G$ is $k$-triangular if each of its edge is contained in at least $k$ triangles. It is conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow. A triangle-path in a graph $G$ is a sequence of distinct triangles $T_1 T_2\cdots T_k$ in $G$ such that for $1\leq i\leq k-1, |E(T_i )\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j>i+1$. Two edges $e,e''\in E(G)$ are triangularly connected if there is a triangle-path $T_1,T_2,\cdots, T_k$ in $G$ such that $e\in E(T_1)$ and $e''\in E(T_k)$. Two edges $e,e''\in E(G)$ are equivalent if they are the same, parallel or triangularly connected. It is easy to see that this is an equivalent relation. Each equivalent class is called a triangularly connected component. In this paper, we prove that every 4-edge-connected triangular graph $G$ is ${\mathbb Z}_3$-connected, unless it has a triangularly connected component which is not ${\mathbb Z}_3$-connected but admits a nowhere-zero 3-flow.     

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.     

局部连通图的群Z_3-连通性

本文研究了局部连通图的群连通性的问题.利用不断收缩非平凡Z_3-连通子图的方法,在G是3-边连通且局部连通的无爪无沙漏图的情况下,获得了G不是群Z_3-连通的当且仅当G是K_4或W_5.推广了当G是2-边连通且局部3-边连通时,G是群Z_3-连通的这个结果.     

Cycle Covers of Planar 2-Edge-Connected Graphs

A graph with n vertices is said to have a small cycle cover provided its edges can be covered with at most (2n ? 1)/3 cycles. Bondy [2] has conjectured that every 2-connected graph has a small cycle cover. In [3] Lai and Lai prove Bondy’s conjecture for plane triangulations. In [1] the author extends this result to all planar 3-connected graphs, by proving that they can be covered by at most (n + 1)/2 cycles. In this paper we show that Bondy’s conjecture holds for all planar 2-connected graphs. We also show that all planar 2-edge-connected graphs can be covered by at most (3n ? 3)/4 cycles and we show an infinite family of graphs for which this bound is attained.     

Minimum cycle coverings and integer flows

It was conjectured by Fan that if a graph G = (V,E) has a nowhere-zero 3-flow, then G can be covered by two even subgraphs of total size at most |V| + |E| - 3. This conjecture is proved in this paper. It is also proved in this paper that the optimum solution of the Chinese postman problem and the solution of minimum cycle covering problem are equivalent for any graph admitting a nowhere-zero 4-flow.     

Decomposing Graphs into Long Paths

It is known that the edge set of a 2-edge-connected 3-regular graph can be decomposed into paths of length 3. W. Li asked whether the edge set of every 2-edge-connected graph can be decomposed into paths of length at least 3. The graphs C ₃, C ₄, C ₅, and K ₄−e have no such decompositions. We construct an infinite sequence {F _i } _i=0 ^∞ of nondecomposable graphs. On the other hand, we prove that every other 2-edge-connected graph has a desired decomposition. This revised version was published online in June 2006 with corrections to the Cover Date.     

Small Edge Sets Meeting all Triangles of a Graph

It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for ‘triangle-3-colorable’ graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K ₄-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.     

On 3-Edge-Connected Supereulerian Graphs

The supereulerian graph problem, raised by Boesch et al. (J Graph Theory 1:79–84, 1977), asks when a graph has a spanning eulerian subgraph. Pulleyblank showed that such a decision problem, even when restricted to planar graphs, is NP-complete. Jaeger and Catlin independently showed that every 4-edge-connected graph has a spanning eulerian subgraph. In 1992, Zhan showed that every 3-edge-connected, essentially 7-edge-connected graph has a spanning eulerian subgraph. It was conjectured in 1995 that every 3-edge-connected, essentially 5-edge-connected graph has a spanning eulerian subgraph. In this paper, we show that if G is a 3-edge-connected, essentially 4-edge-connected graph and if for every pair of adjacent vertices u and v, d _G (u) + d _G (v) ≥ 9, then G has a spanning eulerian subgraph.     

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.     

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.