多项式可压缩性与禁止的有向森林
Polynomial Compressibility and Forbidden Oriented Forests
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中文总结 AI 辅助
该研究构造了有向图反例，反驳了有向团数有界时τ-有界性的多项式猜想，并证明禁止图产生多项式τ-有界类仅当基础图为森林，同时给出若干具体界。
中文摘要 AI 辅助
对于一个非空的无环有向图$H$，设$p(H)$为最长有向路径的阶数，并设$\tau(H)$为最小的正整数$n$，使得$H$同态到每个$n$阶锦标赛。对于所有$p\ge3$和$g\ge1$，我们构造了一个连通的无环有向图$H$，其基础图的围长大于$g$，绝对和相对有向团数均等于三，并且\\[ p(H)=p,\qquad \tau(H)=r_{\mathrm{tr}}(p), \\] 其中$r_{\mathrm{tr}}(p)=2^{\Theta(p)}$是传递$p$顶点锦标赛的锦标赛拉姆齐数。这反驳了在绝对或相对有向团数有界的情况下关于多项式界的猜想。它还表明，一个禁止图只有在基础图是森林时才能产生多项式$\tau$-有界类。对于固定的$g$，这些例子的最小阶数以$p$的多项式为界。一个单独的构造给出了最大入度和出度$O(p^2)$，且对$g$一致。对于$p=4$，最小阶数为$2^{\Theta(g)}$。我们还建立了四个顶点树的两个方向的每个方向的多项式$\tau$-有界性。纯爪形情形由已知的$O(p^4)$界得出。对于$p\ge2$，我们得到了混合爪形和$P_4$的单转向方向上的界$2p-2$，以及$P_4$的有向和交替方向上的界$4$和$3p-2$。在交替情形下，当基础图是无三角形时，$\tau(H)=p(H)$。
英文摘要
For a nonempty acyclic oriented graph $H$, let $p(H)$ be the order of a longest directed path and let $τ(H)$ be the least positive integer $n$ such that $H$ admits a homomorphism to every tournament of order $n$. For all $p\ge3$ and $g\ge1$, we construct a connected acyclic oriented graph $H$ with underlying girth greater than $g$, absolute and relative oriented clique numbers equal to three, and \[ p(H)=p,\qquad τ(H)=r_{\mathrm{tr}}(p), \] where $r_{\mathrm{tr}}(p)=2^{Θ(p)}$ is the tournament Ramsey number for a transitive $p$-vertex tournament. This disproves the conjectured polynomial bounds under bounded absolute or relative oriented clique number. It also shows that a forbidden graph can yield a polynomially $τ$-bounded class only if its underlying graph is a forest. For fixed $g$, the least order of these examples is bounded by a polynomial in $p$. A separate construction gives maximum in- and outdegree $O(p^2)$, uniformly in $g$. For $p=4$, the least order is $2^{Θ(g)}$. We also establish polynomial $τ$-boundedness for every orientation of the two four-vertex trees. The pure-claw case follows from the known $O(p^4)$ bound. We obtain the bound $2p-2$ for mixed claws and one-turn orientations of $P_4$ when $p\ge2$, and bounds $4$ and $3p-2$ for the directed and alternating orientations of $P_4$, respectively. In the alternating case, $τ(H)=p(H)$ when the underlying graph is triangle-free.
发表机构
- School of Science, Shenyang Aerospace University（沈阳航空航天大学理学院）
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