Many flows in the group connectivity setting Matt DeVos∗ Rikke Langhede† Bojan Mohar‡ Robert S´amalˇ § May 21, 2020 Abstract Two well-known results in the world of nowhere-zero flows are Jaeger’s 4-flow theorem asserting that every 4-edge-connected graph has a nowhere-zero Z2 × Z2-flow and Seymour’s 6-flow theorem asserting that every 2-edge-connected graph has a nowhere-zero Z6-flow. Dvoˇr´ak and the last two authors of this paper extended these results by proving the existence of exponentially many nowhere-zero flows under the same assumptions. We revisit this setting and provide extensions and simpler proofs of these results. The concept of a nowhere-zero flow was extended in a significant paper of Jaeger, Linial, Payan, and Tarsi to a choosability-type setting. For a fixed abelian group Γ, an oriented graph G =(V,E) is called Γ-connected if for every function f : E → Γ there is a flow φ : E → Γ with φ(e) 6= f(e) for every e ∈ E (note that taking f = 0 forces φ to be nowhere-zero). Jaeger et al. proved that every oriented 3-edge- connected graph is Γ-connected whenever |Γ|≥ 6. We prove that there are exponentially many solutions whenever |Γ|≥ 8. For the group Z6 we prove that for every oriented 3-edge-connected G = (V,E) with √ℓ/ log ℓ ℓ = |E| − |V | ≥ 11 and every f : E → Z6, there are at least 2 flows φ with φ(e) 6= f(e) for every e ∈ E. Keywords: nowhere-zero flow; group connectivity; counting MSC: 05C21; 05C30 1 Introduction arXiv:2005.09767v1 [math.CO] 19 May 2020 Throughout this paper we permit graphs to have loops and parallel edges. We use standard graph theory terminology and notation as in [3] and [4]. In particular, if G is a graph and X V (G) and S E(G), we ⊆ ⊆ ∗Department of Mathematics, Simon Fraser University, Burnaby, B.C. V5A 1S6. E-mail: [email protected]. Supported by an NSERC Discovery Grant (Canada) †Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Lyngby, Denmark. E-mail: [email protected]. ‡Department of Mathematics, Simon Fraser University, Burnaby, B.C. V5A 1S6. E-mail: [email protected]. Supported in part by the NSERC Discovery Grant R611450 (Canada) and by the Research Program P1-297 of ARRS (Slovenia). §Computer Science Institute (CSI) of Charles University, Malostransk´en´amˇest´ı25, 118 00 Prague, Czech Republic. E-mail: [email protected]ff.cuni.cz. Partially supported by grant 19-21082S of the Czech Science Foundation. This project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 823748. 1 2 let G X and G S denote the subgraph of G obtained by removing all vertices in X (and their incident − − edges), and removing all edges in S (but keeping all vertices), respectively. We also write G[X] to denote the subgraph induced on X. For a graph G = (V, E), we define a k-coloring to be a function f : V 1, 2,...,k with the property →{ } that f(u) = f(v) for every uv E. If G is equipped with an orientation and v V , we let δ−(v) denote the 6 ∈ ∈ set of edges that have v as terminal vertex and δ+(v) the set of edges with v as initial vertex; we also put + δ(v)= δ (v) δ−(v). If Γ is an additive abelian group, a function φ : E Γ is called a flow or a Γ-flow if ∪ → the following rule is satisfied at every v V : ∈ φ(e)= φ(e). e δ+(v) e δ−(v) ∈X ∈X We say that φ is nowhere-zero if 0 φ(E). IfΓ= Z and φ(e) < k for every e E we call φ a k-flow. Note 6∈ | | ∈ that if φ is a flow and we reverse the direction of an edge e, we may replace φ(e) with φ(e) and this gives − us a flow relative to this new orientation. Since this operation preserves the properties of nowhere-zero and k-flow, the presence of a nowhere-zero Γ-flow or a nowhere-zero k-flow depends only on the underlying graph and not on the particular orientation. The study of nowhere-zero flows was initiated by Tutte [13] who observed that these are dual to colorings in planar graphs. Namely, he proved the following. Theorem 1.1 (Tutte [13]). Let G and G∗ be dual planar graphs and orient the edges of G arbitrarily. If Γ is an abelian group with Γ = k, the number of k-colorings of G∗ is equal to k times the number of nowhere-zero | | Γ-flows of G. The above theorem has the curious corollary that for planar graphs, the number of nowhere-zero flows in an abelian group Γ depends only on Γ . Tutte proved that this holds more generally for arbitrary graphs. | | That is, for any abelian groups Γ and Γ with Γ = Γ , the number of nowhere-zero Γ -flows is equal 1 2 | 1| | 2| 1 to the number of nowhere-zero Γ2-flows in every oriented graph. Furthermore, the inherent monotonicity in coloring (every graph with a k-coloring has a k′-coloring for every k′ k) is also present in flows. This ≥ follows from another theorem of Tutte asserting that for every k 2, an oriented graph G has a nowhere-zero ≥ k-flow if and only if it has a nowhere-zero Zk-flow. In addition to establishing these fundamental properties, Tutte made three fascinating conjectures concerning nowhere-zero flows that have directed the field since then. Conjecture 1.2 (Tutte [13, 15, 3]). Let G be an oriented 2-edge-connected graph. 1. G has a nowhere-zero 5-flow. 2. If G does not have a Petersen graph minor, it has a nowhere-zero 4-flow. 3. If G is 4-edge-connected, it has a nowhere-zero 3-flow. Despite a wealth of research, all three of Tutte’s conjectures remain open. Below we have summarized some of the most significant results to date on the presence of nowhere-zero flows. 3 Theorem 1.3. Let G be an oriented graph. 1. (Seymour [10]) If G is 2-edge-connected, it has a nowhere-zero 6-flow. 2. (Jaeger [7]) If G is 4-edge-connected, it has a nowhere-zero 4-flow. 3. (Lov´asz, Thomassen, Wu, Zhang [9]) If G is 6-edge-connected, it has a nowhere-zero 3-flow. Beyond showing that a graph G has a k-coloring, one may look to find lower bounds on the number of k-colorings. Although there are infinite families of planar graphs where any two 4-colorings differ by a permutation of the colors (so there are just 4! = 24 in total), for 5-colorings the following theorem provides an exponential lower bound. Theorem 1.4 (Birkhoff, Lewis [2]). Every simple planar graph on n vertices has at least 2n 5-colorings. By Theorem 1.1 the above result implies that 2-edge-connected planar graphs have exponentially many nowhere-zero Z5-flows. More recently, Thomassen [12] showed a similar exponential bound for the number of 3-colorings of triangle-free planar graphs, which yields exponentially many nowhere-zero Z3-flows in any 4- edge-connected planar graph. In a recent work, Dvoˇr´ak and the latter two authors of this paper investigated the problem of finding exponentially many nowhere-zero flows in general graphs and established the following results. Theorem 1.5 (Dvoˇr´ak, Mohar, and S´amalˇ [5]). Let G be an oriented graph with n vertices and m edges. 2(m n)/9 1. If G is 2-edge-connected, it has 2 − nowhere-zero Z6-flows. 2. If G is 4-edge-connected, it has 2n/250 nowhere-zero Z Z -flows. 2 × 2 (n 2)/12 3. If G is 6-edge-connected, it has 2 − nowhere-zero Z3-flows. We have revisited this topic and found shorter proofs and improved exponential bounds in the first two instances appearing in the following theorem. Theorem 1.6. Let G be an oriented graph with n vertices and m edges. (m n)/3 1. If G is 3-edge-connected, it has 2 − nowhere-zero Z6-flows. 2. If G is 4-edge-connected, it has 2n/3 nowhere-zero Z Z -flows. 2 × 2 Theorem 1.6 will be proved in Section 3, see Theorems 3.2 and 3.3. List-coloring provides a more general setting than that of standard coloring. Let G = (V, E) be a graph and let = Lv v V be a family of sets indexed by the vertices of G. An -coloring of G is a function L { } ∈ L f so that f(v) L holds for every v V and f(u) = f(v) whenever uv E. We say that a graph G is ∈ v ∈ 6 ∈ k-choosable if an -coloring exists whenever L k holds for every v V . Note that every k-choosable L | v| ≥ ∈ graph is necessarily k-colorable since we may take L = 1, 2,...,k for every v V . For planar graphs, v { } ∈ Thomassen [11] proved the following theorem showing the existence of exponentially many -colorings when L every set has size five. 4 Theorem 1.7 (Thomassen [11]). Let G = (V, E) be a simple planar graph of order n. If = Lv v V is a L { } ∈ family of sets with L =5 for every v V , then there are at least 2n/9 -colorings of G.