奇数Hadwiger猜想是“几乎”可判定的
The odd Hadwiger's conjecture is "almost'' decidable
摘要：
由Gerads和Seymour在20世纪90年代初提出的奇异Hadwiger猜想是著名的Hadwiger猜想的一个类似物。它说，每个没有奇数$K_t$ -子图是$(t-1)$ -可着色的。已知这个猜想对$t \leq 5$是正确的，但对$t \geq 5$的情况则是完全开放的。到目前为止，最一般的结果是，没有奇数$K_t$ -子图的图是$O(t \sqrt{\log t})$ -可着色的。在本文中，我们解决这个猜想从一个算法的观点，并显示以下内容：对于一个给定的图$G$和任何固定的$t$，有一个多项式时间算法输出以下之一：\begin{enumerate} \item a $(t-1)$-coloring of $G$, or \item an odd $K_{t}$-minor of $G$, or \item after making all "reductions" to $G$, the resulting graph $H$ (which is an odd minor of $G$ and which has no reductions) has a tree-decomposition $(T, Y)$ such that torso of each bag $Y_t$ is either \begin{itemize} \item of size at most $f_1(t) \log n$ for some function $f_1$ of $t$, or \item a graph that has a vertex $X$ of order at most $f_2(t)$ for some function $f_2$ of $t$ such that $Y_t-X$ is bipartite. Moreover, degree of $t$ in $T$ is at most $f_3(t)$ for some function $f_3$ of $t$. \end{itemize} \end{enumerate}让我们观察到，最后一个奇数未成年人$H$确实是一个最小的反例奇数Hadwiger猜想的情况下$t$。所以我们的结果是一个极小反例满足lsat结论。
英文摘要：
The odd Hadwiger's conjecture, made by Gerads and Seymour in early 1990s, is an analogue of the famous Hadwiger's conjecture. It says that every graph with no odd $K_t$-minor is $(t-1)$-colorable. This conjecture is known to be true for $t \leq 5$, but the cases $t \geq 5$ are wide open. So far, the most general result says that every graph with no odd $K_t$-minor is $O(t \sqrt{\log t})$-colorable. In this paper, we tackle this conjecture from an algorithmic view, and show the following: For a given graph $G$ and any fixed $t$, there is a polynomial time algorithm to output one of the following: \begin{enumerate} \item a $(t-1)$-coloring of $G$, or \item an odd $K_{t}$-minor of $G$, or \item after making all "reductions" to $G$, the resulting graph $H$ (which is an odd minor of $G$ and which has no reductions) has a tree-decomposition $(T, Y)$ such that torso of each bag $Y_t$ is either \begin{itemize} \item of size at most $f_1(t) \log n$ for some function $f_1$ of $t$, or \item a graph that has a vertex $X$ of order at most $f_2(t)$ for some function $f_2$ of $t$ such that $Y_t-X$ is bipartite. Moreover, degree of $t$ in $T$ is at most $f_3(t)$ for some function $f_3$ of $t$. \end{itemize} \end{enumerate} Let us observe that the last odd minor $H$ is indeed a minimal counterexample to the odd Hadwiger's conjecture for the case $t$. So our result says that a minimal counterexample satisfies the lsat conclusion.
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。
