Nowhere-zero 3-flow Research Articles

Nowhere-zero 3-flows in semistrong product of graphs

Let G and H be two graphs. The semistrong productG•H is the graph with vertex set V(G•H)=V(G)×V(H) and edge set E(G•H)={(u1,v1)(u2,v2)|u1u2∈E(G) and v1v2∈E(H) or u1=u2 and v1v2∈E(H)}. It is proved in this paper that if G and H are two nontrivial connected simple graphs, then G•H admits a nowhere-zero 3-flow. This result extends the study of nowhere-zero flows on product graphs by Imrich and Škrekovski, by Shu and Zhang, by Rollová and Škoviera, and by others.

Fulkerson-covers of hypohamiltonian graphs

In this paper, a cycle is a graph in which each vertex has even degree. A Fulkerson-cover of a graph G is a set of six cycles such that each edge of G is in exactly four of the cycles. The well-known Fulkerson-Conjecture asserts that every bridgeless graph has a Fulkerson-cover. We prove that a bridgeless graph G has a Fulkerson cover if and only if there are two disjoint sets E1 and E2 of edges such that E1∪E2 is a cycle and G∖Ei has a nowhere-zero 4-flow for each i=1,2. Using this, together with a result of Jaeger related to crossing numbers, we verify the Fulkerson Conjecture for several known classes of hypohamiltonian graphs in the literatures, including the one based on flip-flops introduced by Chvátal.

Remarks on nowhere-zero flows in signed cubic graphs

We characterise signed cubic graphs that admit a nowhere-zero 3-flow, a nowhere-zero 4-flow, and nowhere-zero flows with values in abelian groups of order 3 and 4. Most of our characterisations feature the concept of an antibalanced signature, one that is switching-equivalent to the all-negative signature. In particular, we prove that a signed cubic graph has a nowhere-zero 3-flow if and only if it has a perfect matching and is antibalanced. Our results suggest several interesting problems for further investigation.

Vector Flows and Integer Flows

A vector ${S}^d$-flow is a flow whose flow values are vectors in $S^d$, where $S^d$ is the set of all unit vectors in $\mathbb{R}^{d+1}$. Jain [Open Problem Garden, http://www.openproblemgarden.org/op/unit_vector_flows (2007)] and Thomassen [J. Combin. Theory Ser. B., 108 (2014), pp. 81--91] proved that a graph has a vector $S^1$-flow if it has a nowhere-zero integer 3-flow. Thomassen [J. Combin. Theory Ser. B., 108 (2014), pp. 81--91] pointed out that a graph admitting a vector $S^1$-flow may not necessarily admit a nowhere-zero integer 3-flow and presented a family of examples showing that the converse is not true. The rank of a vector $S^1$-flow $( D,\bm {f} )$ is defined as the rank of linear space generated by all balanced vectors ${\bm{\epsilon}}(v)=(\epsilon_1(v), \epsilon_2(v), \ldots, \epsilon_b(v))$ for all $v \in V(G)$, where $\epsilon_i(v)$ is the difference between the number of outgoing edges with flow value ${\bm \alpha}_i$ from $v$ and the number of ingoing edges with the same flow value t...

Intersecting 1-factors and nowhere-zero 5-flows

Let G be a bridgeless cubic graph, and μ 2(G) the minimum number k such that two 1-factors of G intersect in k edges. A cyclically n-edge-connected cubic graph G has a nowhere-zero 5-flow if (1) n≥6 and μ 2(G)≤2 or (2) if n≥5μ 2(G)−3.

Parity Subgraphs with Few Common Edges and Nowhere-Zero 5-Flow

A parity subgraph of a graph is a spanning subgraph such that the degrees of all vertices have the same parity in both the subgraph and the original graph. Let $$G$$G be a cyclically 6-edge-connected cubic graph. Steffen (Intersecting 1-factors and nowhere-zero 5-flows 1306.5645, 2013) proved that $$G$$G has a nowhere-zero 5-flow if $$G$$G has two perfect matchings with at most two intersections. In this paper, we show that $$G$$G has a nowhere-zero 5-flow if $$G$$G has two parity subgraphs with at most two common edges, which generalizes Steffen's result.

Nowhere-zero 3-flows of claw-free graphs

Let A denote an abelian group and G be a graph. If a graph G∗ is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say G can be A-reduced to G∗. A graph is claw-free if it has no induced subgraph K1,3. Let N1,1,0 denote the graph obtained from a triangle by adding two edges at two distinct vertices of the triangle, respectively. In this paper, we prove that if G is a simple 2-connected {claw,N1,1,0}-free graph, then G does not admit nowhere-zero 3-flow if and only if G can be Z3-reduced to two families of well characterized graphs or G is one of the five specified graphs.

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.

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. Nanasiova and M. Skoviera [J. Algebraic Combin., 2009, 30: 103–110].

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 Signed Graphs

Tutte observed that every nowhere-zero $k$-flow on a plane graph gives rise to a $k$-vertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph $G$ has a face-$k$-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero $k$-flow. However, if the surface is nonorientable, then a face-$k$-coloring corresponds to a nowhere-zero $k$-flow in a signed graph arising from $G$. Graphs embedded in orientable surfaces are therefore a special case that the corresponding signs are all positive. In this paper, we prove that if an 8-edge-connected signed graph admits a nowhere-zero integer flow, then it has a nowhere-zero 3-flow. Our result extends Thomassen's 3-flow theorem on 8-edge-connected graphs to the family of all 8-edge-connected signed graphs. And it also improves Zhu's 3-flow theorem on 11-edge-connected signed graphs.

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 Tutteʼs 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-connected, Jaegerʼs circular flow conjecture (1984) that for every odd natural number k⩾3, every (2k−2)-edge-connected graph has a modulo k-orientation, etc. It was proved recently by Thomassen that, for every odd number k⩾3, every (2k2+k)-edge-connected graph G has a modulo k-orientation; and every 8-edge-connected graph G is Z3-connected and admits therefore a nowhere-zero 3-flow. In the present paper, Thomassenʼs method is refined to prove the following: For every odd numberk⩾3, every(3k−3)-edge-connected graph has a modulo k-orientation. As a special case of the main result, every 6-edge-connected graph isZ3-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 3-flows of graphs with prescribed sizes of odd edge cuts

The relation between sizes of odd-cycles and chromatic numbers of graphs has been extensively studied. It was conjectured by Bollobás and Erdős (1990) [4] and proved by Gyárfás [A. Gyárfás, Graph with k odd cycle lengths, Discrete Math. 103 (1992) 41–48] that if a graphGhaskdifferent odd cycle sizes, then the chromatic number ofG, χ(G)≤2k+2. Recently, Camacho and Schiermeyer [S.M. Camacho, I. Schiermeyer, Colorings of graphs with two consecutive odd cycle lengths, Discrete Math. 309 (2009) 4916–4919] proved that every graph containing only odd cycles of lengths2k−1and2k+1is 4-colorable. Wang [S.S. Wang, Structure and coloring of graphs with only small odd cycles, SIAM J. Discrete Math. 22 (2008) 1040–1072] characterizes all 3-colorable graphs whose only odd cycles are 3-cycles and 5-cycles; and Kaiser et al. [T. Kaiser, O. Rucký, R. Škrekovski, Graphs with odd cycle lengths 5 and 7 are 3-colorable, SIAM J. Discrete Math. 25 (2012) 1069–1088] proved that every graph with odd cycle lengths 5 and 7 is 3-colorable.In this paper, we investigate the dual version of this problem: the relation between integer flows and odd edge cuts (the integer flow problem is a dual version of the vertex coloring problem, and the dual of an odd-cycle is an odd-edge-cut). The main result in this paper shows that, for each odd integer k≥3, if the size of every odd edge cut of a graph G is in {k,k+2,…,3k−2}, then G admits a nowhere-zero 3-flow if and only if G cannot be reduced to three well-defined families of graphs.

Contractible configurations on 3-flows in graphs satisfying the Fan-condition

Let G be a 2-edge-connected graph of order n such that each pair of vertices x and y at distance 2 satisfies max{d(x),d(y)}>n2 (weak Fan-condition, [G.H. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory Ser. B 37 (1984) 221–227]). It is obtained in this paper that G admits a nowhere-zero 3-flow and is a 3-flow contractible configuration except for a family of well-characterized counterexamples. This result generalizes some earlier results about 3-flows, 3-flow contractible configurations under some degree-sum conditions ([G. Fan, C. Zhou, Ore condition and nowhere-zero 3-flows, SIAM J. Discrete Math. 22 (2008) 288–294], [R. Luo, R. Xu, J.H. Yin, G.X. Yu, Ore-condition and Z3-connectivity, European J. Combin. 29 (2008) 1587–1595]).

Nowhere-zero 3-flows in matroid base graph

The base graph of a simple matroid M = (E,B) is the graph G such that V (G) = B and E(G) = {BB′: B,B′ ∈ B, |B / B′| = 1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connected simple matroid M is Z3-connected if |V (G)| ⩽ 5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if |V (G)| = 4. Furthermore, if for every connected component Ei (i ⩽ 2) of M, the matroid base graph Gi of Mi = M|Ei has |V (Gi)| ⩽ 5, then G is Z3-connected which also implies that G admits nowhere-zero 3-flow immediately.

Relating edge-coverings to the classification of [formula omitted]-magic graphs

Let G=(V,E) be a finite graph and let (A,+) be an abelian group with identity 0. Then G is A-magic if and only if there exists a function ϕ from E into A−{0} such that for some c∈A, ∑e∈E(v)ϕ(e)=c for every v∈V, where E(v) is the set of edges incident to v. Additionally, G is zero-sum A-magic if and only if ϕ exists such that c=0. In this paper, we explore Z2k-magic graphs in terms of even edge-coverings, graph parity, factorability, and nowhere-zero 4-flows. We prove that the minimum k such that bridgeless G is zero-sum Z2k-magic is equal to the minimum number of even subgraphs that cover the edges of G, known to be at most 3. We also show that bridgeless G is zero-sum Z2k-magic for all k≥2 if and only if G has a nowhere-zero 4-flow, and that G is zero-sum Z2k-magic for all k≥2 if G is Hamiltonian, bridgeless planar, or isomorphic to a bridgeless complete multipartite graph. Finally, we establish equivalent conditions for graphs of even order with bridges to be Z2k-magic for all k≥4.

Nowhere-zero 3-flows and [formula omitted]-connectivity in bipartite graphs

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. Let F12 be a family of graphs such that G∈F12 if and only if G is a simple bipartite graph on 12 vertices and δ(G)=4. Let G be a simple bipartite graph on n vertices. It is proved in this paper that if δ(G)≥⌈n4⌉+1, then G admits a nowhere-zero 3-flow with only one exceptional graph. Moreover, if G∉F12 with the minimum degree at least ⌈n4⌉+1 is Z3-connected. The bound is best possible in the sense that the lower bound for the minimum degree cannot be decreased.

Flows and parity subgraphs of graphs with large odd-edge-connectivity

The odd-edge-connectivity of a graph G is the size of the smallest odd edge cut of G. Tutte conjectured that every odd-5-edge-connected graph admits a nowhere-zero 3-flow. As a weak version of this famous conjecture, Jaeger conjectured that there is an integer k such that every k-edge-connected graph admits a nowhere-zero 3-flow. Jaeger [F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B 26 (1979) 205–216] proved that every 4-edge-connected graph admits a nowhere-zero 4-flow. Galluccio and Goddyn [A. Galluccio, L.A. Goddyn, The circular flow number of a 6-edge-connected graph is less than four, Combinatorica 22 (2002) 455–459] proved that the flow index of every 6-edge-connected graph is strictly less than 4. This result is further strengthened in this paper that the flow index of every odd-7-edge-connected graph is strictly less than 4. The second main result in this paper solves an open problem that every odd-(2k+1)-edge-connected graph contains k edge-disjoint parity subgraphs. The third main theorem of this paper proves that if the odd-edge-connectivity of a graph G is at least4⌈log2|V(G)|⌉+1, then G admits a nowhere-zero 3-flow. This result is a partial result to the weak 3-flow conjecture by Jaeger and improves an earlier result by Lai et al. The fourth main result of this paper proves that every odd-(4t+1)-edge-connected graph G has a circular(2t+1)even subgraph double cover. This result generalizes an earlier result of Jaeger.

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.

Nowhere-zero flows in Cartesian bundles of graphs

We extend the results of Imrich and Škrekovski [J. Graph Theory 43 (2003) 93–98] concerning nowhere-zero flows in Cartesian product graphs to ‘twisted’ Cartesian products, that is, Cartesian bundles. Our main result states that every Cartesian bundle of two graphs without isolated vertices has a nowhere-zero 4-flow.