Snarks and Flow-Critical Graphs 1 CˆandidaNunes da Silva a Lissa Pesci a Cl´audioL. Lucchesi b a DComp – CCTS – ufscar – Sorocaba, sp, Brazil b Faculty of Computing – facom-ufms – Campo Grande, ms, Brazil Abstract It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutte’s Flow Conjectures. Keywords: nowhere-zero k-flows, Tutte’s Flow Conjectures, 3-edge-colouring, flow-critical graphs. 1 Nowhere-Zero Flows Let k > 1 be an integer, let G be a graph, let D be an orientation of G and let ϕ be a weight function that associates to each edge of G a positive integer in the set {1, 2, . , k − 1}. The pair (D, ϕ) is a (nowhere-zero) k-flow of G 1 Support by fapesp, capes and cnpq if every vertex v of G is balanced, i. e., the sum of the weights of all edges leaving v equals the sum of the weights of all edges entering v. Tutte [15] defined the concept of k-flows as a generalization of the concept of k-face-colourings after observing that, for any planar graph, a k-flow can be obtained from a k-face-colouring and vice-versa. The concept of k-flow is the most general of the two since the fact that a graph admits or not a k-flow does not depend on it being embedable in the plane or any other surface, while admitting a face k-colouring does depend on a particular embedding of the graph. Tutte then proposed three celebrated conjectures regarding k- flows of graphs, known as the Five-, Four- and Three-Flow Conjectures, that generalize to non-planar graphs three famous theorems on face colouring of planar graphs. These conjectures are still open and appear in the Unsolved Problems section of the new book by Bondy and Murty [1, Open Problems 95, 96, 97]. According to the authors, these conjectures are “arguably the most significant problems in the whole of graph theory”. Only two of these Conjectures are of interest in this paper: the Five- and Four-Flow Conjectures. Tutte’s Five-Flow Conjecture [15] asserts that every 2-edge-connected graph has a 5-flow. The Four-Flow Conjecture [16] asserts that every 2-edge-connected graph without a Petersen minor has a 4-flow. A famous theorem of Tutte (see [1, Theorem 21.11]), stated below, plays a fundamental role in this research area. Theorem 1.1 (Tutte) A cubic graph admits a 4-flow if and only if it admits a 3-edge-colouring. Seymour [14] proved that every 2-edge-connected graph has a 6-flow. Also, Robertson, Sanders, Seymour and Thomas 2 proved the Four-Flow Conjecture for cubic graphs, by stating the following theorem: Theorem 1.2 Every 2-edge-connected cubic graph without a Petersen minor admits a 3-edge-colouring. 2 Snarks In order to understand the structure of 2-edge-connected graphs that do not admit a 3-edge-colouring, researchers came to the definition of a class of graphs called snarks. The definition of snarks aims not only at distinguishing the cubic graphs that do not admit a 3-edge-colouring, but also at specifying the 2 The proof consists of five papers by various subsets of these authors, of which only paper [13] has been published so far. minimal such graphs, in some sense. A snark is then formally defined as a cubic graph that is cyclically 4-edge-connected, has girth at least five and does not admit a 3-edge-colouring. A graph is cyclically k-edge-connected if it does not contain an edge-cut on less than k edges whose removal yields at least two connected components containing cycles. The restriction to cyclically 4-edge- connected graphs with girth at least five in the definition comes from the fact that any non-3-edge-colourable cubic graph that does not satisfy one of these two properties can be reduced to a smaller cubic non-3-edge-colourable graph by a sequence of edge removals and contractions (see [17, Lemma 7.3.10]). We therefore derive the following result. Theorem 2.1 Every 2-edge-connected cubic graph that does not have a 3- edge-colouring has a snark as a minor. The first snark discovered was the Petersen graph and it was presented in 1898 by Petersen as the smallest 2-edge-connected cubic graph without a 3-edge-colouring. For several decades, very few other snarks were found. This apparent difficulty in finding such graphs inspired Descartes [7] to christen them snarks after the Lewis Carroll poem ‘The Hunting of the Snark’. Al- though Descartes was the first to use the term snark (see [1, page 462]), most references in the literature attribute the name choice to Gardner [11], who made the term widely known. Good reports on the delightful history behind snarks hunt and their relevance in Graph Theory can be found in [1,2,17]. In 1975, Isaacs [12] showed that there were infinite families of snarks, one of such being the flower-snarks. Recently, a computational approach to snark hunt has been explored by Brinkmann et al. [10], who developed a computer program to generate all snarks on up to 36 vertices, all of which were recently made available at the House of Graphs Database [9]. In the study of both the Five- and Four-Flow Conjectures, snarks play an important role. First observe that, according to Theorem 2.1, the snarks are essentially the only ones that do not admit a 4-flow. Also, the subclass of cubic graphs is the only one for which the Four-Flow Conjecture has been proved. Therefore, a good understanding of the structure of snarks may be helpful in finding an alternative proof for this particular case. For the Five-Flow Conjecture, it is well known that every smallest counter- example must be cubic. Moreover, it cannot admit a 4-flow, whence, by The- orems 1.1 and 2.1, any smallest counter-example must be a snark. Therefore, a good understanding of the structure of snarks may help to resolve the Five- Flow Conjecture. In this paper, we take a new approach to the study of the structure of snarks, by trying to identify the minimal snarks, in some sense. Our concept of minimality is defined in the next section. 3 Critical Graphs Let k > 1 be an integer. A graph G is k-colour-critical if it does not admit a k- vertex-colouring but every proper subgraph of G admits a k-vertex-colouring. Dirac [8] defined the concept of critical graphs for the vertex-colouring problem and discovered several properties of such graphs. Likewise, in a previous paper [4] we defined the concept of (edge-)k-flow-critical graphs. A graph G is k-flow-critical if it does not admit a k-flow but the graph G/e obtained from G by the contraction of an arbitrary edge e does admit a k-flow. Any smallest counter-example to either the Five-Flow or the Four-Flow Conjectures must be 5-flow-critical and 4-flow-critical graphs, respectively. Any 2-edge-connected 5-flow-critical graph is a counter-example to the Five-Flow Conjecture and no such graph is known. On the other hand, candidates to 4-flow-critical graphs are abundant: no snark has a 4-flow. Evidently, no snark is a counter- example to the Four-Flow Conjecture as, by Theorem 1.2, all of them have a Petersen minor. In contrast, any smallest counter-example to the Five-Flow Conjecture must be a snark. Moreover, it must be a non-4-flow-critical snark since every 2-edge-connected 4-flow-critical graph admits a 5-flow, as shown in [5, Theorem 3.12]. These observations lead us to investigate which snarks are 4-flow-critical. We did so by writing a computer program (see [3]) to check whether a graph is 4-flow-critical. The program runs in exponential time and simply checks, for every possible combination of orientation and weight assignments (D, ϕ) to the edges of a graph G, if the following properties are satisfied: (i) all vertices are balanced; (ii) all but exactly two vertices are balanced. Graph G is then 4-flow-critical if condition (i) is not satisfied for any (D, ϕ) and there is a (D, ϕ) leaving precisely the ends of edge e unbalanced, for every edge e of G. We first applied our program to several famous snarks, namely the Pe- tersen, Blanuˇsa,Loupekhine, Celmins-Swart, double-star and Szekeres snarks, as well as the flower-snarks with up to 60 vertices (see [2] for pictures). All these snarks are 4-flow-critical. We later observed that all flower-snarks are 4-flow-critical, as shown in [6]. Mystified with the fact that all snarks seemed order 10 12 14 16 18 20 22 24 26 28 snarks 1 0 0 0 2 6 20 38 280 2900 4-flow-critical snarks 1 0 0 0 2 1 2 0 111 33 Table 1 Number of 4-flow-critical snarks of order at most 28 to be 4-flow-critical, we decided to check all snarks with up to 28 vertices 3 , and then discovered several non-4-flow-critical snarks.