Petersen Cores and the Oddness of Cubic Graphs

Abstract

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let E i be the set of edges contained in precisely i members of the k 1-factors. Let µ k (G) be the smallest |E 0 | over all lists of k 1-factors of G. We study lists by three 1-factors, and call

Cores, joins and the Fano-flow conjectures

“…The following results were already obtained in [4], but now it turns out that they are just a particular case of our previous theorem.…”

An equivalent formulation of the Fan-Raspaud Conjecture and related problems

“…However, many others are still open, such as Conjecture 2.1 proposed independently by Berge and Fulkerson in the 1970s as well, and Conjecture 2.2 by Fan and Raspaud (see [10] and [7], respectively). These two conjectures are related to the behaviour of the union and intersection of sets of perfect matchings, and properties of this kind are already largely studied: see, amongst others, [1,2,15,16,17,19,22,23,25,30,31]. In this paper we prove that a seemingly stronger version of the Fan-Raspaud Conjecture is equivalent to the classical formulation (Theorem 3.3), and so to another interesting formulation proposed in [21] (see also [18]).…”

The smallest nontrivial snarks of oddness 4

“…Again, there is a striking difference between the trivial and the nontrivial snarks. Most cyclically 4-edge-connected snarks in Table 3 have µ 3 = 6, which is the minimal possible value for snarks with ω = 4, according to [15,Corollary 2.4]. By contrast, most trivial snarks from Table 4 have µ 3 > 6.…”

“…The values of the remaining invariants vary over the set M. It is quite remarkable that the automorphism group of every graph in M is a 2-group (or is trivial). The second group of invariants comprises those which are of particular interest for snarks: perfect matching index π, resistance ρ, weak oddness ω , and two invariants introduced in [15,26,27] and denoted by µ 3 and γ 2 ; see also [5] for a recent survey. The perfect matching index of a bridgeless cubic graph G, denoted by π(G) (also known as excessive index and denoted by χ e ), is the smallest number of perfect matchings that cover all the edges of G [2,7].…”

“…The oddness of a bridgeless cubic graph G is the smallest number of odd circuits in a 2-factor of G, and the resistance of G is the smallest number of vertices (or edges) of G whose removal yields a 3-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].…”