In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups.

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.

Let G be a bridgeless cubic graph. The resistance of G, denoted r(G), is the minimum number of edges which can be removed from G in order to render 3-edge-colourability. The oddness of G, denoted ω(G), is the minimum number of odd components in a 2-factor of G. The colouring defect of G (or simply, the defect of G), denoted µ3(G), is the minimum number of edges not contained in any set of three perfect matchings of G. These three parameters are regarded as measurements of uncolourability of snarks, partly because any one of these parameters equal zero if and only if G is 3-edge-colourable. It is also known that r(G) ≥ ω(G) and that µ3(G) ≥ 3 2 ω(G) [5,6]. We have shown that the ratio of oddness to resistance can be arbitrarily large for non-trivial snarks [1]. It has also been shown that the ratio of the defect to oddness can be arbitrarily large for non-trivial snarks, although this result was only shown for graphs with oddness equal to 2 [7]. In the same paper, the question was posed whether there exists non-trivial snarks for given resistance r or given oddness ω, and arbitrarily large defect. In this paper, we prove a stronger result: For any positive integers r ≥ 2, even ω ≥ r, and d ≥ 3 2 ω, there exists a non-trivial snark G with r(G) = r, ω(G) = ω and µ3(G) ≥ d.

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