论文标题
关于双刺和两个块循环的细分的评论
Remarks on the subdivisions of bispindles and two-blocks cycles in highly chromatic digraphs
论文作者
论文摘要
a $（2+1）$ - bispindle $ b（k_1，k_2; k_3）$是两个$ xy $ -dipath的结合，各个长度的$ k_1 $和$ k_2 $，一个$ yx $ - 长度$ k_3 $的dipaths，所有这些dipaths is bepthes bepliwes is beplise by nossilly binwise binwise binwise in nossilly binwise in Binwise binwise binwise billy inthin in osthiph in ofthipt。最近，Cohen等。推测，对于每个积极整数$ k_1，k_2，k_3 $，都有一个整数$ g（k_1，k_2，k_3）$，以至于每个牢固连接的digraph都不包含$ b（k_1，k_2; k_3）$的$ b（k_1，k_2; k_3）$在$ g（k_1，k_2; k_3）中只有$ g（k_1，k_1，k_1，k_1，k_1，k_1，k_1，k_1，k_1，k_2）， $ k_2 = 1 $。对于Hamiltonian Digraphs，我们证明了Cohen等人的猜想，即$ g（k_1，k_2，k_3）\ leq 4k $，其中$ k = max \ {k_1，k_2，k_2，k_3 \} $。一个两个块周期$ c（k_1，k_2）$是两个内部不连接$ xy $ - 长度$ k_1 $和$ k_2 $的联合。 Addario等。当被问及不包含两个块周期$ c（k_1，k_2）$的不包含分区的强挖掘的色数是否可以通过$ o（k_1+k_2）$限制，这仍然是一个开放的问题。假设$ k = max \ {k_1，k_2 \} $，最佳到达的上限是Kim等人，为$ 12K^2 $。在本文中，我们猜测该界限可以稍微提高到$ 4K^2 $，我们确认了某些特定情况的猜想。此外，我们为Addario等人的问题提供了一个积极的答案，该类别的digraphs具有Hamiltonian指向路径。
A $(2+1)$-bispindle $B(k_1,k_2;k_3)$ is the union of two $xy$-dipaths of respective lengths $k_1$ and $k_2$, and one $yx$-dipath of length $k_3$, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. conjectured that, for every positive integers $k_1, k_2, k_3$, there is an integer $g(k_1, k_2, k_3)$ such that every strongly connected digraph not containing subdivisions of $B(k_1, k_2; k_3)$ has a chromatic number at most $g(k_1, k_2, k_3)$, and they proved it only for the case where $k_2=1$. For Hamiltonian digraphs, we prove Cohen et al.'s conjecture, namely $g(k_1, k_2, k_3)\leq 4k$, where $k=max\{k_1, k_2, k_3\}$. A two-blocks cycle $C(k_1,k_2)$ is the union of two internally disjoint $xy$-dipaths of length $k_1$ and $k_2$ respectively. Addario et al. asked if the chromatic number of strong digraphs not containing subdivisions of a two-blocks cycle $C(k_1,k_2)$ can be bounded from above by $O(k_1+k_2)$, which remains an open problem. Assuming that $k=max\{k_1,k_2\}$, the best reached upper bound, found by Kim et al., is $12k^2$. In this article, we conjecture that this bound can be slightly improved to $4k^2$ and we confirm our conjecture for some particular cases. Moreover, we provide a positive answer to Addario et al.'s question for the class of digraphs having a Hamiltonian directed path.
