snark 的阻力、奇度与着色缺陷
Resistance, oddness and colouring defect of snarks
AI总结：
本文研究非平凡 snark 的阻力、奇度与着色缺陷关系，构造性证明在给定 $r\ge2$、偶数 $ω\ge r$ 及任意大的 $d\ge\frac32ω$ 时，存在满足相应参数的图。
摘要：
设 $G$ 是一个无桥三次图。$G$ 的阻力（resistance）记为 $r(G)$，是指为使其成为三边可着色图而可从 $G$ 中删除的最少边数。$G$ 的奇度（oddness）记为 $ω(G)$，是指 $G$ 的一个二因子中奇数分量的最少个数。$G$ 的着色缺陷（colouring defect，或简称缺陷）记为 $μ_3(G)$，是指不包含在 $G$ 的任意三个完美匹配集合中的最少边数。这三个参数被视为衡量 snark 不可着色性的指标，部分原因在于这些参数中的任意一个等于零，当且仅当 $G$ 是三边可着色的。此外已知 $r(G) \geq ω(G)$，且 $μ_3(G) \geq \frac{3}{2}ω(G)$。我们已经证明，对于非平凡 snark，奇度与阻力之比可以任意大。也有结果表明，对于非平凡 snark，缺陷与奇度之比可以任意大，尽管该结果只针对奇度等于 2 的图得到证明。在同一篇论文中，作者提出了一个问题：对于给定阻力 $r$ 或给定奇度 $ω$，是否存在缺陷任意大的非平凡 snark。本文证明了一个更强的结果：对于任意正整数 $r \geq 2$、偶数 $ω\geq r$ 以及 $d \geq \frac{3}{2}ω$，存在一个非平凡 snark $G$，满足 $r(G)=r$、$ω(G)=ω$ 且 $μ_3(G) \geq d$。
英文摘要：
Let $G$ be a bridgeless cubic graph. The \textit{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 \textit{oddness} of $G$, denoted $ω(G)$, is the minimum number of odd components in a 2-factor of $G$. The \textit{colouring defect} of $G$ (or simply, the \textit{defect} of $G$), denoted $μ_3(G)$, 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 $r(G) \geq ω(G)$ and that $μ_3(G) \geq \frac{3}{2}ω(G)$ \cite{fiol,jinsteffen}. We have shown that the ratio of oddness to resistance can be arbitrarily large for non-trivial snarks \cite{allie1}. 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 \cite{karabasetal}. 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 $r \geq 2$, even $ω\geq r$, and $d \geq \frac{3}{2}ω$, there exists a non-trivial snark $G$ with $r(G)=r$, $ω(G)=ω$ and $μ_3(G) \geq d$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
