缺陷边着色的奇围长界
Odd-Girth Bounds for Defective Edge Coloring
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中文总结 AI 辅助
该研究针对无环非二部多重图的缺陷边着色问题，证明了结合奇围长参数的上界公式，验证了几类特殊情形下的已知结论，构造了达到等号的几乎全环多重图，并推导了缺陷Goldberg--Seymour猜想的成立范围。
中文摘要 AI 辅助
无环多重图$G$的$(k,d)$-边着色是指使用至多$k$种颜色的边着色，使得每个颜色类构成的子图的最大度不超过$d$。满足条件的最小$k$记为$χ'_d(G)$。设$G$是最大度为$Δ(G)$、奇围长为$g_0(G)$的无环非二部多重图，且$d\ge1$为奇数。我们证明：\\[ χ'_d(G)\le\left\lceil\frac{g_0(G)Δ(G)-1}{dg_0(G)-1}\right\rceil. \\]当$d=1$时，这就是Goldberg对Shannon定理的奇围长改进；当$g_0(G)=3$时，这就是Aboulker、Aubian和Huang的缺陷Shannon界。对于每个奇数$d>1$、每个奇数$g_0\ge3$以及每个$Δ>d$，几乎全环多重图$R(Δ,g_0)$——即边重数在$\lfloorΔ/2\rfloor$和$\lceilΔ/2\rceil$之间交替的奇环，仅两条相邻边的重数为$\lfloorΔ/2\rfloor$——达到等号。我们还推导了缺陷Goldberg--Seymour猜想成立的一个范围。
英文摘要
A $(k,d)$-edge coloring of a loopless multigraph $G$ is an edge coloring using at most $k$ colors such that the subgraph formed by each color class has maximum degree at most $d$. The least such $k$ is denoted by $χ'_d(G)$. Let $G$ be a loopless non-bipartite multigraph with maximum degree $Δ(G)$ and odd girth $g_0(G)$, and let $d\ge1$ be odd. We prove that \[ χ'_d(G)\le\left\lceil\frac{g_0(G)Δ(G)-1}{dg_0(G)-1}\right\rceil. \] For $d=1$, this is Goldberg's odd-girth refinement of Shannon's theorem, while for $g_0(G)=3$ it is the defective Shannon bound of Aboulker, Aubian, and Huang. For every odd $d>1$, every odd $g_0\ge3$, and every $Δ>d$, an almost full ring multigraph $R(Δ,g_0)$, an odd cycle with edge multiplicities alternating between $\lfloorΔ/2\rfloor$ and $\lceilΔ/2\rceil$, except that two consecutive edges have multiplicity $\lfloorΔ/2\rfloor$, attains equality. We also derive a range in which the defective Goldberg--Seymour conjecture holds.
