AI 中文总结
本文将图的k-退化概念扩展到拟阵，定义极小k-退化拟阵并刻画其极端情况，同时扩展拟阵荫度界，证明k-退化拟阵荫度不超过k并给出更精确界。
AI 中文摘要
若图的每个子图都存在度数不超过k的顶点，则称该图为k-退化的。我们将这一概念扩展到拟阵，定义无环拟阵M为k-退化的，当且仅当M的每个限制都包含大小不超过k的余回路；若M的余围长为k，且其每个真限制的余围长都不超过k-1，则称M为极小k-退化的。我们的主要结果刻画了极端极小k-退化拟阵，还扩展了已知的拟阵荫度界，证明k-退化拟阵的荫度不超过k，并提供了更精确的界。
英文摘要
A graph is $k$-degenerate if every subgraph has a vertex of degree at most $k$. We extend this notion to matroids, defining a loopless matroid $M$ to be $k$-degenerate if every restriction of $M$ contains a cocircuit of size at most $k$; $M$ is minimally $k$-degenerate if it has cogirth $k$ and every proper restriction of $M$ has cogirth at most $k-1$. Our main result characterizes extremal minimally $k$-degenerate matroids. We also extend the known arboricity bound for matroids, showing that $k$-degenerate matroids have arboricity at most $k$ and providing sharper bounds.
Comments18 pages
