This paper is a sequel of  where we have proved that the smallest number of vertices of a snark – a connected cubic graph whose edges cannot be properly coloured with three colours – which has cyclic connectivity and oddness at least is . The purpose of the present paper is to show that there are precisely 31 such snarks, all of them having oddness exactly , resistance , and girth . Together with , this paper provides a partial answer to the following question posed in [4, Problem 2], leaving open the existence of cyclically -edge-connected snarks of oddness at least on fewer than 44 vertices:
Problem . Which is the smallest snark (with cyclic connectivity and girth ) of oddness strictly greater than 2?
The oddness of a bridgeless cubic graph is the smallest number of odd circuits in a -factor of , and the resistance of is the smallest number of vertices (or edges) of whose removal yields a -edge-colourable graph. Both invariants are important measures of uncolourability of cubic graphs and have been investigated by numerous authors [1, 5, 11, 12, 13, 15, 25]. One of the reasons why these invariants have recently received so much attention resides in the fact that snarks with large resistance or oddness may provide potential counterexamples to several profound conjectures such as the cycle double cover conjecture, the 5-flow conjecture, and others [11, 13, 14].
The set of all snarks of order 44, cyclic connectivity , and oddness has been constructed with the help of a computer. A detailed description of its members appears in Section 3 where we also give a computer-free proof that each of them has oddness at least . The equality can be established easily by specifying a -factor containing four odd circuits, which can be checked directly.
The snarks constituting the set fall into six classes depending on their structure. A remarkable feature of the set is that all of them are built up from subgraphs of the Petersen graph and a small number of additional vertices. The verification that is a complete set of snarks of order 44, cyclic connectivity , and oddness at least combines purely mathematical considerations with extensive computations; the proof can be found in Section 4. Its mathematical part is essentially a mixture of edge-colouring and cyclic connectivity arguments. In the computational part we perform an operation that takes two cyclically -edge-connected snarks and of order at most 36 and creates from them – in all possible ways – a cubic graph by either removing two adjacent vertices or two nonadjacent edges, and connecting the resulting -valent vertices in with those in . In total, more than graphs have been constructed and checked for oddness 4. The entire computational effort for this project amounts to 25 CPU years.
In Section 5 we analyse a sample of 887 152 cyclically -edge-connected snarks of oddness whose orders range from 46 to 52, as well as 872 snarks of oddness with lower connectivity constructed in . We evaluate various invariants for them such as resistance, perfect matching index, circumference, and others. The purpose of this investigation is to provide grounds for possible prediction of certain properties that snarks with higher oddness might have in general.
We conclude this paper with several open problems. At the end we append the adjacency lists of all 31 snarks constituting the set .
This section collects the most basic definitions and notation needed for understanding the present paper. For a more detailed introduction to the topic we refer the reader to our preceding paper .
2.1. Graphs and multipoles. All graphs in this paper are finite and for the most part simple. However, for the sake of completeness we have to permit graphs containing multiple edges or loops, although these features will be usually excluded by the imposed connectivity or colouring restrictions. For a graph and a subgraph we let denote the number of vertices of , and the subgraph of induced by the vertex set of .
Throughout this paper we use multipoles as a convenient tool for constructing graphs. Every edge of a multipole has two ends and each end can, but need not, be incident with a vertex. An edge which has one end incident with a vertex and the other not is called a dangling edge, and if neither end of an edge is incident with a vertex, it is called an isolated edge. An end of an edge that is not incident with a vertex is called a semiedge. A multipole with semiedges is called a -pole. Two semiedges and of a multipole can be joined to produce an edge connecting the end-vertices of the corresponding dangling edges. Given two -poles and with semiedges and , respectively, we define their complete junction to be the graph obtained by performing the junctions for each . A partial junction is defined in a similar way except that a proper subset of semiedges of is joined to semiedges of . Partial junctions can be used to construct larger multipoles from smaller ones. In either case, whenever a junction of two multipoles is to be performed, we assume that their semiedges are assigned a fixed linear order.
Semiedges in multipoles are often grouped into pairwise disjoint sets, called connectors. The size of a connector is the number of its semiedges. A connector of size is often referred to as an -connector. An -pole is a multipole with semiedges which are distributed into connectors such that the connector is of size . A multipole with two connectors is also called a dipole.
Let and be two dipoles with connectors , and , , respectively. If , and each of these two connectors is endowed with a linear order, we can construct a new dipole
with connectors and by performing the junctions of the semiedges from with those with respect to the corresponding orderings. The resulting dipole is called the junction of and .
2.2. Cyclic connectivity. Let be a connected graph. An edge-cut of a graph is any set of edges of such that is disconnected. For example, if a proper subset of vertices or induced subgraph of , then the set of all edges with exactly one end in is an edge-cut in . An edge-cut is said to be cycle-separating if at least two components of contain cycles. We say that a connected graph is cyclically -edge-connected if no set of fewer than edges is cycle-separating in . Let denote the cycle rank of . The cyclic connectivity of , denoted by , is the largest number for which is cyclically -connected (cf. [20, 22]). It is not difficult to see that if and only if any two curcuits of have a vertex in common. For cubic graphs this can happen only when is the complete bipartite graph , the complete graph on four vertices, or the graph consisting of two vertices and three parallel edges joining them.
For a cubic graph with , the value coincides with the usual vertex-connectivity and edge-connectivity of , so cyclic connectivity provides a natural extension of the classical connectivity parameters for cubic graphs. Another useful observation is that the value of cyclic connectivity remains invariant under subdivisions and adjoining new vertices of degree . In particular, homeomorphic graphs have the same value of cyclic connectivity.
For edge-cuts that separate an acyclic component from the rest of the graph we have the following easy but useful observation.
A connected acyclic -pole has vertices.
2.3. Edge-colourings. A -edge-colouring of a graph is a mapping such that adjacent edges receive distinct colours; the same definition applies to multipoles. A graph or a multipole which admits a -edge-colouring will be called colourable, otherwise it will be called uncolourable. A -connected uncolourable cubic graph is called a snark. A snark is nontrivial if it is cyclically -edge-connected and has girth at least .
In the study of colourings of cubic graphs it is often convenient to take the colours to be the non-zero elements of the group , because in this case 3-edge-colourings correspond to nowhere-zero -flows. We identify the colours , , and with , , and , respectively.
The following well known lemma – in fact, an immediate consequence of flow continuity – is a fundamental tool in the study of snarks.
(Parity Lemma) Let be a -pole endowed with a proper -edge-colouring with colours , , and . If the set of all semiedges of contains edges of colour for , then
In this paper we study snarks that are far from being -edge-colourable. Two measures of uncolourability are relevant for this paper. The oddness of a bridgeless cubic graph is the smallest number of odd circuits in a 2-factor of . The resistance of a cubic graph is the smallest number of vertices of which have to be removed in order to obtain a colourable graph. Somewhat surprisingly, the required number of vertices to be deleted is the same as the number of edges that have to be deleted in order to get a -edge-colourable graph (see [24, Theorem 2.7]). In fact, in many cases it is more convenient to delete edges rather than vertices.
Obviously, if is colourable, then . If is uncolourable, then both and . Observe that for every bridgeless cubic graph we have since deleting one edge from each odd circuit in a -factor leaves a colourable graph. On the other hand, the Parity Lemma implies that never equals , which together with a standard Kempe chain recolouring argument yields that if and only if [24, Lemma 2.5]. The difference between and can be arbitrarily large in general , nevertheless, resistance can serve as a convenient lower bound for oddness because it is somewhat easier to handle.
By the Parity Lemma, every colouring of a -pole has one of the following types: , , , and (for a precise definition of the type of a colouring see ). Observe that every colourable -pole admits at least two different types of colourings. Indeed, we can start with any colouring and switch the colours along an arbitrary Kempe chain to obtain a colouring of another type. Colourable -poles thus can have two, three, or four different types of colourings. Those attaining exactly two types are particularly important for the study of snarks; we call them colour-open -poles, as opposed to colour-closed multipoles discussed in more detail in .
There are two types of colour-open -poles. A -pole will be called isochromatic if its semiedges can be partitioned into two pairs such that in every colouring of the semiedges within each pair receive the same colour. A -pole will be called heterochromatic of its semiedges can be partitioned into two pairs such that in every colouring of the semiedges within each pair receive distinct colours. Typical examples of isochromatic and heterochromatic -poles are depicted in Figure 2 and Figure 3, respectively.
The adjectives “isochromatic” and “heterochromatic” can be similarly applied to graphs: a subgraph of a cubic graph will be called isochromatic if attaching a dangling edge to every -valent vertex produces an isochromatic -pole; a heterochromatic subgraph is defined similarly.
3 The 31 snarks
Let denote the set of all cyclically 4-edge-connected snarks with at most 36 vertices; as mentioned in , the set consists of nonisomorphic graphs. For any two snarks and from let us apply the following operation:
From each form a -pole by either removing two adjacent vertices or two nonadjacent edges and by retaining the dangling edges.
Construct a cubic graph by identifying the semiedges of with those of after possibly applying a permutation to the semiedges of or .
The resulting graph will be called a -join of and . Define to be the set all pairwise nonisomorphic cyclically -edge-connected snarks of order with oddness at least that can be expressed as a -join of two not necessarily distinct graphs from .
We have implemented a program which applies a -join in all possible ways to any two input graphs; see  for more details concerning the program. We have applied this program in all possible ways to every pair of graphs from that lead to a graph on 44 vertices, and then tested which of the constructed graphs have oddness at least 4. This computation took approximately 2 CPU years on a cluster consisting of Intel Xeon E5-2660 CPU’s at 2.60GHz and produced snarks with oddness exactly .
The set consists of exactly snarks, each of them having oddness exactly and girth .
In the remainder of this section we describe the 31 snarks in detail and provide a computer-free proof that each of them has oddness at least . Their adjacency lists, displayed in the order as they were generated, can be found in Appendix. Graph number 28 is illustrated in Figure 1, two more of the 31 graphs are depicted in Figure 1 and Figure 4 in our previous paper  which represent graphs number 15 and 17, respectively.
The basic building blocks of all snarks constituting the set are five multipoles , , , , and , described below, all of them arising from the Petersen graph by removing vertices or severing edges. Every individual member of may have a small number of additional vertices not belonging to any of these subgraphs.
Let denote the -pole arising from the Petersen graph by removing two adjacent vertices and grouping the semiedges formerly incident with the same vertex to the same connector; it is shown in Figure 2. Since every -edge-colouring of assigns the edges in the same connector the same colour, is an is an isochromatic -pole. It is the only isochromatic -pole on eight vertices and at the same time the smallest connected isochromatic -pole. In the symbolic representation of we represent the edges in one of the connectors by bold lines, see Figure 6. This clearly determines the other connector as well. Due to the symmetry of the Petersen graph, the two connectors of are interchangeable.
We emphasise that the connectors of , as well as those of the other four building blocks , , , and , are unordered. Different orderings are needed for the construction of the members of , a fact which partially explains a relatively large size of this set.
Let denote a -pole formed from the Petersen graph by severing two independent edges and grouping the semiedges arising from the same edge to the same connector. Every -edge-colouring of assigns different colours to the dangling edges within the same connector, so is a heterochromatic -pole. There are two ways how to select a pair of independent edges in the Petersen graph – either at distance or at distance . Accordingly, there exist two nonisomorphic heterochromatic -poles on ten vertices, denoted by and , respectively. Observe that there exists no bridgeless heterochromatic -pole with fewer vertices. The two heterochromatic -poles on ten vertices are displayed in Figure 3. In the symbolic representation of we distinguish the two connectors of again by using bold lines for one of the connectors, see Figure 6. Again, the connectors of both and are interchangeable.
Let denote the -pole obtained from the Petersen graph by removing an arbitrary vertex and severing an edge not incident with ; the semiedges formerly incident with are put to one connector and those arising by severing are put to the other connector. The resulting -pole is shown in Figure 4. Given an arbitrary -edge-colouring of , the edges of the -connector receive two distinct colours and , while the edges of the -connector receive the colours , , , where is any colour from . In the symbolic representation of the edges of the -connector are drawn bold, see Figure 6.
Let denote the -pole arising from the Petersen graph by removing a path of length . The two -connectors consist of the edges formerly incident with the same end-vertex of the path, the -connector gets the remaining edge. The resulting -pole is shown in Figure 5. The important property of consists in the fact that every -edge-colouring of assigns the edges of one of the -connectors two distinct colours and while the edges of the other -connector receive the same colour ; the fifth edge of is coloured . In the symbolic representation of the edges of one -connector are drawn bold and the -connector edge is drawn dashed, see Figure 6. As with and before, the two -connectors of are interchangeable.
We divide the 31 snarks of into six classes depending on the number of disjoint copies of , , , and , and on the number of additional vertices in the graph. For example, by we denote the set of all snarks from that consist of two copies of , which need not be isomorphic, two copies of , one copy of , and one additional vertex. We do not distinguish between the two varieties and of because both of them play the same structural role within the snark in question and their contribution to increasing the oddness is the same. The six classes of are
In the rest of this section we describe each of the classes in detail and prove that every member has . In our discussion we will be employing certain standard combinations of the multipoles , , and defined by means of junctions. The order of semiedges in connectors is in all cases irrelevant. We define the -poles and , the -pole , where the junction involves the -connector of , and the -pole defined as follows: in the -pole subdivide one of the edges between the two copies of and subsequently attach a dangling edge to the new vertex of degree ; the connectors of are defined in the obvious way. The multipoles , , , and are illustrated in Figure 7(d).
The following statements hold true:
Observe that is uncolourable because it is a junction of an isochromatic -pole with a heterochromatic -pole . Hence, . Similarly, every colouring of assigns its -connector two distinct colours while is isochromatic, so is uncolourable too, and therefore again. If the -pole was colourable, then every -edge-colouring of would assign the edges adjacent to the dangling edge of the -connector two distinct colours. At most one of these colours would match the colour of the edge connecting the two copies of in . Therefore, the isochromatic property of at least one copy of in would always be violated. Hence, is uncolourable, and therefore .
Finally we prove that . Since is contained in and , we infer that as well. In particular, the graph obtained from by identifying the semiedges within each connector is a snark. To prove that suppose to the contrary that , and let be a vertex such that is -edge-colourable. Clearly, cannot belong to a copy of for otherwise would still contain a copy of and therefore would be uncolourable. Thus must belong to the copy of , and hence every -edge-colouring of must assign the same colour to both semiedges in any of the connectors. Now we can match the semiedges of each connector, thereby obtaining a -edge-colouring of . Since is a snark, such a colouring does not exist. This contradiction proves that .
To establish the required equalities for each particular multipole one has to display the corresponding colourings. Finding such colourings is straightforward, and therefore is left to the reader. ∎
In our analysis of the snarks we will often need to distinguish between different copies of the same basic building block . For this purpose we will be using upper indices, for example , , etc.
Class 1: 2h + 3i (7 graphs)
This class splits into two subclasses, Class 1a and Class 1b, depending on how the building blocks are connected between each other. Both subclasses are illustrated in Figure 8(b). Each graph from Class 1 contains a copy of and a copy of . Since the copies of and are disjoint, Lemma 3 implies that and hence .
Class 1a consists of graphs 15, 17, and 18, while Class 1b contains graphs 1, 4, 21, and 24.
Class 2: 2h + 2i + N + 1 (4 graphs)
The structure of graphs from Class 2 is illustrated in Figure 9. It can be seen that every graph from Class 2 contains two disjoint copies of , therefore . Note each of the induced subgraphs , , , and is uncolourable, the last of them being isomorphic to . Thus if there exist two vertices and in such that is colourable, then either one of them lies in and the other lies in , or one of them lies in and the other lies in . In all other cases one of the mentioned uncolourable subgraphs will remain intact. Without loss of generality we may assume that belongs to and belongs . Since is isochromatic, the edges between and have the same colour. Because is heterochromatic, the colours the edges between and are different. The inverting property of ensures that the edges between and also have the same colour. If we now identify the dangling edges within each connector of , we obtain a colouring of the Petersen graph with one vertex removed. This contradiction proves that that , which means that , as required.
Class 2 contains graphs 10, 11, 19, and 27.
Class 3: H + 4i + 2 (6 graphs)
This class splits into three subclasses, 3a, 3b, and 3c. Their structure is represented in Figure 10. Every graph from Class 3a and Class 3b contains a copy of and a copy , which are disjoint, therefore and . Class 3c is somewhat different and requires a separate argument. Consider an arbitrary graph graph from Class 3c. Since contains a copy of , we have . Note that each of the subgraphs , , and has resistance 1. Thus if there exist two vertices and in such that is -edge-colourable, then one of them, say , lies in and the other belongs to . Since and remain intact in , the isochromatic property of implies that the edge connecting to has the same colour as the edge connecting to . Since is heterochromatic, we conclude that lies in . As a consequence, remains intact in too, which means that the two edges and joining to are equally coloured as well. But then the induced colouring of yields a -edge-colouring of the Petersen graph with a single vertex removed. This contradiction proves that and .
Class 3a contains graphs 5 and 6, Class 3b contains graphs 7 and 23, and Class 3c contains graphs 9 and 26.
Class 4: H + 3i + T + 1 (10 graphs)
Class 4 has two subclasses 4a and 4b, both shown in Figure 11(b). Every graph from Class 4 contains disjoint copies of and , so and .
Class 4a consists of graphs 2, 3, 13, and 14, and Class 4b contains graphs 8, 12, 16, 20, 22, and 25.
Class 5: 5i + 4 (2 graphs)
Class 5 has two subclasses Class 5a and Class 5b; they are represented in Figure 12(b). Observe that every member of Class 5 contains a -pole isomorphic to , namely , whose resistance is 1 by Lemma 3. Hence, every member of Class 5 is indeed a snark.
Now, let us consider a graph from Class 5a. To prove that suppose the contrary. Then , which means that there exist vertices and in such that is colourable. Since each of the -poles , with , is isomorphic to , and , none of them survives in . It follows that one of and lies in and the other one lies in . Let be a -edge-colouring of . Without loss of generality we may assume that , and . Since remains intact in , we have . Similarly, since is intact, we have . Consider the colours of and . Since remains intact , we deduce that and hence . Furthermore, because otherwise the induced colouring of would give rise to a -edge-colouring of the Petersen graph with one vertex removed, which is impossible. Therefore or , but since we conclude that . However, the isochromatic property of forces , which is in conflict with the value because and are adjacent. Therefore and .
Next, let us consider a graph from Class 5b. We wish to prove that . Suppose the contrary. Then , which again means that there exist vertices and in such that is -edge-colourable. In this case it is easy to see that one of the vertices belongs and the other one belongs to . Suppose that the vertex lies in so that remains intact. Take an arbitrary -edge-colouring of , and let . Since and are also left intact in , we deduce that , , and finally . However, the edges and are adjacent, so this is impossible. Therefore , which means that is left intact. In this situation we can derive a contradiction in a similar way as for Class 5a. This proves that for every graph from Class 5b.
Both Class 5a and Class 5b consist of a single graph each – graphs 28 and 30, respectively.
Class 6: 4i + T + 3 (2 graphs)
The structure of Class 6 is displayed in Figure 13. Let us first observe that every member of Class 6 is uncolourable because it contains the subgraph isomorphic to , whose resistance equals 1 by Lemma 3. It follows that . To prove that suppose to the contrary that . Since each of the subgraphs , with , is also uncolourable, there exist vertices and such that is -edge-colourable. Let be one such colouring. Without loss of generality we may assume that , , and . Since and are left intact, we have and . Put together, the edges of the -connector of receive three distinct colours from the colouring . As previously mentioned, every -edge-colouring of forces a repeated colour in the -connector. Therefore must belong to . It follows that remains intact and therefore . But then the induced colouring of yields a -edge-colouring of the Petersen graph with a single vertex removed. This contradiction proves that .
Class 6 contains two nonisomorphic graphs – 29 and 31.
Properties of graphs in
We have determined the values of several invariants for the 31 snarks in . In most cases the computations were performed by a computer. The evaluated invariants can be divided into two groups. The first group is constituted by general invariants: namely, the order of the automorphism group, genus (minimum genus of an orientable surface upon which a given graph can be drawn without intersections), diameter, radius, and circumference (the maximal circuit length in a graph). The values of these invariants for individual members of are summarised in Table 1. In particular, all members of have circumference , that is, where is the number of vertices. The values of the remaining invariants vary over the set . It is quite remarkable that the automorphism group of every graph in is a -group (or is trivial).
The second group of invariants comprises those which are of particular interest for snarks: perfect matching index , resistance , weak oddness , and two invariants introduced in [15, 26, 27] and denoted by and ; see also  for a recent survey. The perfect matching index of a bridgeless cubic graph , denoted by (also known as excessive index and denoted by ), is the smallest number of perfect matchings that cover all the edges of [2, 7]. The weak oddness of a cubic graph , denoted by , is the smallest number of components of odd order in an even factor of ; by an even factor we mean a spanning subgraph with all degrees even. Given a bridgeless cubic graph , we define to be the smallest number of common edges that two perfect matchings of can have, and let be the smallest number of edges of that are left uncovered by the union of any three perfect matchings of .
The perfect matching index of is bounded below by and equals if and only if is -edge-colourable. It is believed, by a conjecture of Berge (see ), that for every bridgeless cubic graph . This conjecture, if true, thus divides all snarks into two subclasses, those with perfect matching index equal to , and those with perfect matching index . We have determined that for every .
The remaining invariants, along with oddness, can be regarded as measures of uncolourability as they take value on -edge-colourable graphs, and positive values otherwise. Their comparison with oddness and resistance may therefore be very instructive.
First of all, using a computer we have determined that for all snarks . As regards weak oddness, it is easy to see that is an even integer such that . Furthermore, if , then necessarily as well, because otherwise would immediately yield whence . In particular, for all snarks we have . It is known, however, that in general both the difference and can be arbitrarily large (see [1, 17]).
We finish this section by discussing the previously mentioned invariants and . In [27, Proposition 2.1] Steffen proved that for every bridgeless cubic graph . Using a computer we have determined that for every , which shows that every snark from fulfils the upper bound on set by with equality. Similarly, in [15, Corollary 2.4] Jin and Steffen proved that for every bridgeless cubic graph. Using a computer we have determined that all snarks have , which means that they again reach the upper bound on in terms of with equality. Snarks with the latter property have a very special structure of sets of edges left uncovered by three perfect matchings and therefore deserve special attention.
The values of perfect matching index, resistance, weak oddness, , and for the snarks of are summarised in Table 2.
Each of the six classes described above gives rise to an infinite family of snarks with oddness at least and cyclic connectivity . It is sufficient to replace the basic building blocks , , , and obtained from the Petersen graph with similar structures created from any cyclically -edge-connected snark. With a little additional care one can construct infinite families of snarks with increasing oddness.
4 Completeness of
In this section we prove that the set , constructed and analysed in Section 3, is the complete set of pairwise nonisomorphic snarks with cyclic connectivity , oddness at least , and minimum order. Our point of departure is the following theorem proved in .
The smallest number of vertices of a snark with cyclic connectivity and oddness at least is . The girth of each such snark is at least .
This result is a consequence of the following stronger and more detailed result from  which will be needed for the proof of the main result of this paper.
Let be a snark with oddness at least , cyclic connectivity , and minimum number of vertices. Let be a cycle-separating -edge-cut in whose removal leaves components and . Then, up to permutation of the index set , exactly one of the following occurs.
Both and are uncolourable, in which case each of them can be extended to a cyclically -edge-connected snark by adding two vertices.
is uncolourable and is heterochromatic, in which case can be extended to a cyclically -edge-connected snark by adding two vertices, and can be extended to a cyclically -edge-connected snark by adding two isolated edges.
is uncolourable and is isochromatic, in which case can be extended to a cyclically -edge-connected snark by adding two vertices, and can be extended to a cyclically -edge-connected snark by adding two vertices, except possibly . In the latter case, is a partial junction of two colour-open -poles, which may be isochromatic or heterochromatic in any combination.
Here is our main result:
The set is the complete set of snarks with cyclic connectivity and oddness at least on vertices.
Let be the set of all snarks with cyclic connectivity and oddness at least on vertices. To prove Theorem 4 it suffices to show that . The general strategy of the proof is to show that every snark is a -join of two cyclically -edge-connected snarks of order at most 36. As soon as this is done, one can perform -joins in all possible ways that give rise to a cyclically -edge-connected snark of order , identify those whose oddness equals , and check whether all of them belong to .
In order to apply this strategy we employ Theorem 4. It implies that we can split each into two subgraphs and each of which can be extended to cyclically a -edge-connected snark by adding at most two vertices. The difficult part of the proof arises when one of the subgraphs, namely , has 36 vertices and is an isochromatic -pole on vertices (see (iii) of Theorem 4). Adding two adjacent vertices to is now useless because the list of all cyclically -edge-connected snarks is known only up to 36 vertices [4, 8]. Instead, we show that it is possible to add two isolated edges to in such a way that a cyclically -edge-connected snark of order 36 is created. A detailed analysis that precedes this step is the core of the proof of Theorem 4, which now follows.
Let be a snark with cyclic connectivity 4 and oddness at least 4 on 44 vertices. We wish to prove that . Suppose the contrary. In order to derive a contradiction we first establish four claims.
Claim 1. Every cycle-separating -edge-cut of determines two components, one uncolourable on and one isochromatic on vertices. Both components are -edge-connected.
Proof of Claim 1. Let be an arbitrary cycle-separating -edge-cut in and let and be the components of . Since is a minimum cycle-separating edge-cut, both and are easily seen to be -edge-connected. ¿From Theorem 4 we deduce that one of the components, say , is uncolourable. If was either uncolourable or heterochromatic, then it would have at least 10 vertices and hence has at most vertices. Using Theorem 4 again we could conclude that both and can be extended to snarks of order at most , so would be a -join of two snarks from and therefore a member of . This contradiction shows that is isochromatic.
Now we prove that the isochromatic component produced by an arbitrary cycle-separating -edge-cut has only eight vertices. Suppose to the contrary that there exists a cycle separating edge-cut in such that the isochromatic component, denoted by , has at least ten vertices. Choose to minimise the number of vertices of . Clearly, the other component of has at most vertices and, by the first part of the proof, it is uncolourable. It follows that also has at least ten vertices, so has at most vertices too. Theorem 4 further implies that can be extended to a cyclically -edge-connected snark by adding two vertices. If , the same theorem implies that can also be extended to a cyclically -edge-connected snark by adding two vertices. Since both and have order at most 36, belongs to – a contradiction. Therefore has a -edge-cut. By Theorem 4 (iii), is a partial junction of two colour-open -poles and . Since both and are cycle-separating -edge-cuts, both and must be isochromatic. If any of them had more than eight vertices, then the corresponding edge-cut would contradict the choice of . Therefore is a partial junction of two copies of the isochromatic -pole on eight vertices, so has 16 vertices and has 28 vertices. It follows that can be expressed in the form where is the Petersen graph, is a snark on 30 vertices, and denotes a -join of cubic graphs and which employs -poles resulting from the removal of two adjacent vertices from both and , while denotes a -join of cubic graphs and which employs a -pole obtained from by removing two nonadjacent edges and a -pole obtained from by removing two adjacent vertices. Using a computer we have constructed all graphs arising in this way and verified that each of them either belongs to or has oddness at most . This contradiction establishes Claim 1.
Claim 2. Every -edge-cut in separates a subgraph with at most eight vertices from the rest of .
Proof of Claim 2. Let be a -edge-cut in . If one of the components of is acyclic, then, by Lemma 2, this component has two vertices. If is cycle-separating, then the conclusion follows from Claim 1. Claim 2 is proved.
Fix a cycle-separating -edge-cut in ; we will refer to as the principal -edge-cut of . As before, let and be the components of where is uncolourable. By Claim 1, the other component is isochromatic on eight vertices. Let be the set of end-vertices of in . Since is independent, we have .
The remainder of the proof is devoted to proving the following fact:
The component can be extended to a cyclically -edge-connected snark by adding two edges between the vertices of .
As a consequence, will be a -join of two graphs from , and therefore a member of . This will provide a final contradiction.
Claim 3. is cyclically -edge-connected and has a cycle-separating -edge-cut.
Proof of Claim 3. If , then adding to two edges joining the vertices of in an arbitrary manner would produce a cyclically -edge-connected snark of order 36. Consequently, would belong to . Therefore . To prove that suppose to the contrary that has a cycle-separating 2-edge-cut , and let and be the two components of . Since is cyclically -edge-connected, two edges of join to and other two edges of join to . Thus is a cycle-separating -edge-cut which separates from . Since contains more then eight vertices, Claim 1 implies that is an isochromatic dipole on eight vertices. Likewise, is an isochromatic dipole on eight vertices. Put together, has altogether vertices, which is again a contradiction. Therefore . Finally, is homeomorphic to a certain cubic graph with on 32 vertices, so is not a subdivision of the complete graph and therefore contains a cycle-separating -edge-cut. This establishes Claim 3.
Before we can proceed we need two definitions. First, a cycle-separating -edge-cut in will be called balanced if each component of is incident with exactly two edges of the principal -edge-cut . Otherwise, will be called unbalanced. Second, let and be two cycle-separating -edge-cuts in . Let and be the components of and let and be the components of . The edge-cuts and in will be called comparable if or for some .
Claim 4. contains two incomparable balanced -edge-cuts.
Proof of Claim 4. Let us first observe that if is an arbitrary unbalanced -edge-cut in , then adding any two edges between the vertices of in an arbitrary manner will produce a cubic graph where has ceased to be an edge-cut. It follows that if every cycle-separating -edge-cut in is unbalanced, then is a cyclically -edge-connected snark and is a -join of with the Petersen graph. Since has vertices, and we have arrived at a contradiction. Therefore must contain a balanced -edge-cut.
If every pair of balanced -edge-cuts in is comparable, we can arrange the balanced -edge-cuts in an increasing linear order, say . Clearly, there is a component of and a component of such that both of them are disjoint from all of . It is easy to see that contains two vertices of while contains the other two, see Figure 14. Thus we can connect the two vertices of to those of by two edges, producing a cyclically -edge-connected snark . Again, is a -join of with the Petersen graph, so belongs to contrary to the assumption. Therefore contains two incomparable balanced -edge-cuts. This proves Claim 4.
In the rest the proof we explore the structure of arising from a pair of incomparable balanced -edge-cuts. Let and be any two incomparable balanced 3-edge-cuts in . Clearly, has two components, say and , and has two components, say and . The definition of comparable edge-cuts readily implies that each of the subgraphs is non-empty. Let be the number of edges between