Weak oddness as an approximation of oddness and resistance in cubic graphs

We introduce weak oddness ωw, a new measure of uncolourability of cubic graphs, defined as the least number of odd components in an even factor. For every bridgeless cubic graph G, %(G) ≤ ωw(G) ≤ ω(G), where %(G) denotes the resistance of G and ω(G) denotes the oddness of G, so this new measure is an approximation of both oddness and resistance. We demonstrate that there are graphs G satisfying %(G) < ωw(G) < ω(G), and that the difference between any two of those three measures can be arbitrarily large. The construction implies that if we replace a vertex of a cubic graph with a triangle, then its oddness can decrease by an arbitrary large amount. 1 Oddness and resistance Cubic graphs naturally fall into two classes depending on whether they do or do not admit a 3-edge-colouring. Besides the trivial family of graphs with bridges, which are trivially uncolourable, there are many examples of 2-edge-connected cubic graphs that do not admit a 3-edge-colouring. Such graphs are called snarks ; sometimes they are required to satisfy additional conditions, such as cyclic 4-edge-connectivity and girth at least five, to avoid triviality. In their many attempts to understand snarks better, researchers have come up with various measures that refine the notion of “being close to 3-edge-colourable”. In Section 2, we introduce a new such measure closely related to oddness and resistance. We follow [7] in our presentation of these two concepts. Every bridgeless cubic graph has a 1-factor [9] and consequently also a 2-factor. It is easy to see that a cubic graph is 3-edge-colourable if and only if it has a 2-factor that only consists of even circuits. In other words, snarks are those cubic graphs which have an odd circuit in every 2-factor. The minimum number of odd circuits in a 2-factor of a bridgeless cubic graph G is its oddness, and is denoted by ω(G). Since every cubic graph has even number of vertices, its oddness must also be even. The relevance of oddness stems from the importance of snarks. The crux of many important problems and conjectures, like the Tutte’s 5-flow conjecture or the cycle double 1 ar X iv :1 60 2. 02 94 9v 1 [ cs .D M ] 9 F eb 2 01 6 cover conjecture, consists in dealing with snarks. While most of these problems are exceedingly difficult for snarks in general, they are often tractable for those that are close to being 3-edge-colourable. For example, the 5-flow conjecture has been verified for snarks with oddness at most 2 (Jaeger [6]) and for cyclically 6-edge-connected snarks with oddness at most 4 [13], and the cycle double cover conjecture has been verified for snarks with oddness at most 4 (Huck and Kochol [4], Häggkvist and McGuinness [3]). Snarks with large oddness thus remain potential counterexamples to these conjectures and therefore deserve further study. Another natural measure of uncolourability of a cubic graph is based on minimising the use of the fourth colour in a 4-edge-colouring of a cubic graph, which can alternatively be viewed as minimising the number of edges that have to be deleted in order to get a 3-edge-colourable graph. Surprisingly, the required number of edges to be deleted is the same as the number of vertices that have to be deleted in order to get a 3-edge-colourable graph (see [11, Theorem 2.7]). This quantity is called the resistance of G, and will be denoted by %(G). Observe that %(G) ≤ ω(G) for every bridgeless cubic graph G since deleting one edge from each odd circuit in a 2-factor leaves a colourable graph. The difference between %(G) and ω(G) can be arbitrarily large in general [12].