On the smallest snarks with oddness 4 and connectivity 2

A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in [Electron. J. Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and remarked and that there are exactly two such graphs of that order. However, this remark is incorrect as -- using an exhaustive computer search -- we show that there are in fact three snarks with oddness 4 on 28 vertices. In this note we present the missing snark and also determine all snarks with oddness 4 up to 34 vertices.


Introduction and main result
The chromatic index of a graph G is the minimum number of colours required for an edge colouring of that graph such that no two adjacent edges have the same colour. It follows from Vizing's classical theorem that a cubic graph has chromatic index either 3 or 4. Isaacs [5] called cubic graphs with chromatic index 3 colourable and those with chromatic index 4 uncolourable. Cubic graphs with bridges can easily be seen to be uncolourable and are therefore considered to be trivially uncolourable.
A snark is a bridgeless cubic graph which is not 3-edge-colourable. Note that in the literature stronger conditions are sometimes required for a graph to be a snark, e.g. that it also must have girth at least 5 and be cyclically 4-edge-connected. (The girth of a graph is the length of its shortest cycle and a graph is cyclically k-edge-connected if the deletion of fewer than k edges from the graph does not create two components both of which contain at least one cycle). Here we will focus on snarks with girth at least 4 since snarks with triangles can be easily reduced to smaller triangle-free snarks.
One of the reasons why snarks are interesting is the fact that the smallest counterexamples to several important conjectures (such as the cycle double cover conjecture [7,8] and the 5-flow conjecture [9]) would be snarks.
The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. The oddness is a natural measure for how far a graph is from being 3-edge-colourable. It is straightforward to see that a bridgeless cubic graph is 3-edge-colourable if and only if it has oddness 0. Also note that the oddness must be even as cubic graphs have an even number of vertices.
Snarks with large oddness are of special interest since several conjectures (including the cycle double cover conjecture and the 5-flow conjecture) are proven to be true for snarks with small oddness.
There is one such snark with cyclic connectivity 2 and one with cyclic connectivity 3.
After the proof of this theorem they remark: "The computer searches referred to in the proof of Theorem 12 can be extended to show that there are exactly two snarks with oddness 4 on 28 vertices -those displayed in Figure 2 (from [6])." However, this remark is incorrect as -using an exhaustive computer search -we have shown that there are in fact three snarks with oddness 4 on 28 vertices, which leads to the following claim.
Claim 2. There are exactly three snarks with oddness 4 on 28 vertices. There are two such snarks with cyclic connectivity 2 and one with cyclic connectivity 3.
The missing snark from Theorem 1 has connectivity 2 and is shown in Figure 1. Using the program snarkhunter [2,3] we have generated all snarks with girth at least 4 up to 34 vertices and tested which of them have oddness greater than 2. The results are listed in Table 1 and in Table 2 for snarks with girth at least 5. In [6,Lemma 2] it was shown that the smallest snarks of a given oddness have girth at least 5.

Figure 1:
The snark with oddness 4 on 28 vertices which was missing in [6].
All snarks with oddness at least 4 up to at least 36 vertices have (cyclic) connectivity 2 or 3, since together with Brinkmann, Hägglund and Markström we have shown [2] that there are no cyclically 4-edge-connected snarks with oddness at least 4 up to at least 36 vertices.
We implemented two independent algorithms to compute the oddness of a bridgeless cubic graph: the first algorithm computes the oddness by constructing perfect matchings while the second algorithm does this by constructing 2-factors directly by searching for disjoint cycles. The source code of both programs can be obtained from [4]. All of our results reported in this article were independently confirmed by both programs.
The graphs from Tables 1 and 2 can be downloaded and inspected in the database of interesting graphs from the House of Graphs [1] by searching for the keywords "snark * oddness 4".
The most symmetric snark with girth at least 4 and with oddness 4 up to 34 vertices is shown in Figure 2. It has 32 vertices, connectivity 2, girth 5 and its automorphism group has order 768. It has 32 vertices and its automorphism group has order 768.