OOn the smallest snarks withoddness 4 and connectivity 2
Jan Goedgebeur ∗ Department of Applied Mathematics, Computer Science and StatisticsGhent UniversityKrijgslaan 281-S9,9000 Ghent, Belgium [email protected]
Submitted: XX; Accepted: XX; Published: XXMathematics Subject Classifications: 05C30, 05C85, 68R10
Abstract 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´aˇcajov´a, Maz´ak and ˇSkoviera showed in [ Electron. J. Combin.
Keywords: cubic graph, snark, chromatic index, oddness, computation, exhaustivesearch
The chromatic index of a graph G is the minimum number of colours required for an edgecolouring of that graph such that no two adjacent edges have the same colour. It followsfrom Vizing’s classical theorem that a cubic graph has chromatic index either 3 or 4.Isaacs [6] called cubic graphs with chromatic index 3 colourable and those with chromaticindex 4 uncolourable . Cubic graphs with bridges can easily be seen to be uncolourableand are therefore considered to be trivially uncolourable. ∗ Supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO). a r X i v : . [ m a t h . C O ] A p r snark is a bridgeless cubic graph which is not 3-edge-colourable. Note that in theliterature stronger conditions are sometimes required for a graph to be a snark, e.g. thatit also must have girth at least 5 and be cyclically 4-edge-connected. (The girth of agraph is the length of its shortest cycle and a graph is cyclically k -edge-connected if thedeletion of fewer than k edges from the graph does not create two components both ofwhich contain at least one cycle). Here we will focus on snarks with girth at least 4 sincesnarks 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 counterex-amples to several important conjectures (such as the cycle double cover conjecture [8, 9]and the 5-flow conjecture [10]) would be snarks.The oddness of a bridgeless cubic graph is the minimum number of odd componentsin any 2-factor of the graph. The oddness is a natural measure for how far a graph isfrom being 3-edge-colourable. It is straightforward to see that a bridgeless cubic graphis 3-edge-colourable if and only if it has oddness 0. Also note that the oddness must beeven as cubic graphs have an even number of vertices.Snarks with large oddness are of special interest since several conjectures (includingthe cycle double cover conjecture and the 5-flow conjecture) are proven to be true forsnarks with small oddness.In [7] Lukot’ka, M´aˇcajov´a, Maz´ak and ˇSkoviera showed the following. Theorem 1 (Theorem 12 in [7]) . The smallest snark with oddness 4 has 28 vertices.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 extendedto show that there are exactly two snarks with oddness 4 on 28 vertices – thosedisplayed in Figure 2 (from [7]).”However, this remark is incorrect as – using an exhaustive computer search – we haveshown that there are in fact three snarks with oddness 4 on 28 vertices, which leads tothe following proposition.
Proposition 2.
There are exactly three snarks with oddness 4 on 28 vertices. There aretwo 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. Usingthe program snarkhunter [2, 3] we have generated all snarks with girth at least 4 up to 34vertices and tested which of them have oddness greater than 2. The results of this searchare listed in Table 1 and in Table 2 for snarks with girth at least 5. In [7, Lemma 2] itwas shown that the smallest snarks of a given oddness have girth at least 5.None of the snarks up to 34 vertices has oddness greater than 4, so this yields thefollowing proposition.
Proposition 3.
The smallest 2-connected snark with oddness at least 6 has at least 36vertices.
Table 1:
The counts of all 2-connected snarks with girth at least 4 up to 34 vertices and thenumber of snarks with oddness 4 among them.
Order All Oddness 4Connectivity 2 Connectivity 3 Total10 1 0 0 012 0 0 0 014 0 0 0 016 0 0 0 018 3 0 0 020 14 0 0 022 107 0 0 024 1 109 0 0 026 15 255 0 0 028 236 966 2 1 330 4 043 956 9 4 1332 74 989 646 33 21 5434 1 500 084 086 139 138 277
Table 2:
The counts of all 2-connected snarks with girth at least 5 up to 34 vertices and thenumber of snarks with oddness 4 among them. igure 1: The snark with oddness 4 on 28 vertices which was missing in [7].
All snarks with oddness at least 4 up to at least 36 vertices have (cyclic) connectivity2 or 3, since it follows from [2] and [5] that there are no cyclically 4-edge-connected snarkswith oddness at least 4 up to at least 36 vertices.We implemented two independent algorithms to compute the oddness of a bridgelesscubic graph: the first algorithm computes the oddness by constructing perfect matchingswhile the second algorithm does this by constructing 2-factors directly by searching fordisjoint cycles. The source code of both programs can be obtained from [4]. All of ourresults 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 ofinteresting 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 verticesis shown in Figure 2. It has 32 vertices, connectivity 2, girth 5 and its automorphismgroup has order 768.
Figure 2:
The most symmetric snark with girth at least 4 and with oddness 4 up to 34 vertices.It has 32 vertices and its automorphism group has order 768. cknowledgements We would like to thank Martin ˇSkoviera for useful suggestions. Most of the computationswere carried out using the Stevin Supercomputer Infrastructure at Ghent University.
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