Abstract
We examine the relationship between two measures of uncolourability of cubic graphs – their resistance and flow resistance. The resistance of a cubic graph $G$, denoted by $r(G)$, is the minimum number of edges whose removal results in a 3-edge-colourable graph. The flow resistance of $G$, denoted by $r_f(G)$, is the minimum number of zeroes in a 4-flow on $G$. Fiol et al. [Electron. J. Combin. 25 (2018), $\#$P4.54] made a conjecture that $r_f(G) \leq r(G)$ for every cubic graph $G$. We disprove this conjecture by presenting a family of cubic graphs $G_n$ of order $34n$, where $n \geq 3$, with resistance $n$ and flow resistance $2n$. For $n\ge 5$ these graphs are nontrivial snarks.
Highlights
Snarks are 2-connected cubic graphs whose edges cannot be properly coloured with three colours
A number of recent results confirm that some of these conjectures become tractable for snarks that are in a certain sense close to 3-edge-colourable graphs, see for example [6, 10, 15]. In this situation it is natural to focus on the study of invariants of cubic graphs that express – in various ways – to what extent a graph differs from a 3-edge-colourable graph
The authors of [4] note that flow resistance can be equivalently defined as the minimum number of edges that have to be contracted in order to obtain a graph that admits a nowhere-zero 4-flow [4, Theorem 33]
Summary
Snarks are 2-connected cubic graphs whose edges cannot be properly coloured with three colours. A similar approach to Tutte’s 3-flow conjecture has recently been taken by DeVos et al [3] where the authors proved that every 3-edge-connected graph admits a 3-flow in which at most one sixth of the edges carries value zero In spite of these facts, flow resistance has so far attracted surprisingly little attention. The authors of [4] note that flow resistance can be equivalently defined as the minimum number of edges that have to be contracted in order to obtain a graph that admits a nowhere-zero 4-flow [4, Theorem 33] They show in [4, Proposition 29] that rf (G) is bounded above by the electronic journal of combinatorics 29(1) (2022), #P1.44 the minimum number of edges that any two perfect matchings of G can have, which is another useful uncolourability measure (denoted in [4] by γ2(G)). If n 5, Gn is cyclically 4-edge-connected and has girth 5
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