Paul Ellis, Daniel T. Soukup
We show that if $D$ is a tournament of arbitrary size then $D$ has finite strong components after reversing a locally finite sequence of cycles. In turn, we prove that any tournament can be covered by two acyclic sets after reversing a locally finite sequence of cycles. This provides a partial solution to a conjecture of S. Thomassé.
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We prove the following conjecture of S. Thomassé: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $ is at most two and each edge is flipped only finitely many times. In addition, we guarantee that in every strong component of $ D^{*} $ all the local edge-connectivities are finite and any edge is reversed at most twice.
Digraphs of Large Girth and Dichromatic Number in Tournaments with Large Dichromatic Number
本文证明每个二色数足够大的竞赛图包含具有大有向围长和大二色数的子有向图，并探讨了用底层图围长替代的情形，该情形蕴含Erdős-Hajnal猜想且对特定竞赛图族成立。
Acyclic Dichromatic Number of Tournaments: these are the Champions
研究竞赛图的无圈双色数，刻画了有足够大无圈双色数的竞赛图中必出现的子竞赛图，证实相关猜想，证明其满足局部到全局性质。
A Simple Construction of Tournaments with Finite and Uncountable Dichromatic Number
本文基于棋盘划分组合引理，给出了二色数恰为任意自然数 $k$ 的竞赛图简单构造，并推广至构造具有不可数二色数且基数为 $\mathbb{R}$ 的完全有向图及有向平衡完全 $n$-部图。
Fractional Dichromatic Number and Domination in Tournaments
本文针对锦标赛的支配数与分数二色数的关系，提出了两个新证明：一个基于归约得到指数界，另一个通过AI启发得到拟线性界。
The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We study oriented graphs, which have dichromatic number more than 2. Such a graph $D$ is $3$-dicritical if the removal of any arc of $D$ reduces the dichromatic number to 2. We construct infinitely many $3$-dicritical oriented graphs. Neumann-Lara found the four $7$-vertex $3$-dichromatic tournaments. We determine the $8$-vertex $3$-dichromatic tournaments, which …
Let $G$ be a graph of order $n$. It is well-known that $α(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $α(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ α(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $α(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $χ(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(Δ^+(D),Δ^-(D))$. We give a short and simple proof for…
We propose a purely combinatorial quadratic time algorithm that for any $n$-vertex $P_{k}$-free tournament $T$, where $P_{k}$ is a directed path of length $k$, finds in $T$ a transitive subset of order $n^{\frac{c}{k\log(k)^{2}}}$. As a byproduct of our method, we obtain subcubic $O(n^{1-\frac{c}{k\log(k)^{2}}})$-approximation algorithm for the optimal acyclic coloring problem on $P_{k}$-free tournaments. Our results are tight up to the $\log(k)$-factor in the following sense: there exist infinite families of $P_{k}$-free tournaments with largest transitive subsets of order at most $n^{\frac{c…
Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers
本文提出模板提升构造，生成同时关于团数和二色数临界的正则顶点传递 Cayley 竞赛图，证明相关猜想并否定有界证书问题，同时刻画循环替换的团数与临界性。
Acyclic dichromatic number of oriented graphs
本文研究了有向图的无环二色数，定义并分析了其与传统二色数的区别，证明了在某些图中无环二色数可能远大于传统二色数，并展示了相关复杂性结果。
We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vecχ}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $\text{mader}_{\vecχ}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is tha…
The dichromatic number $\vecχ(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number…
In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we…
Heroes in oriented complete multipartite graphs
该研究针对有向图的英雄图刻画问题，证明拟传递有向图的英雄图与竞赛图一致，证否了有向多部图相关猜想，并给出有向完全多部图英雄图的近乎完整刻画。
Clique number of tournaments
本文定义有向图的团数为所有反向边图的最小团数，研究其与二色数的关系，并重点探讨竞赛图的dic-有界类，证明H必须具有森林状反向边图，以及dic-有界类满足BIG⇒BIG猜想和多项式dic-有界类满足Erdős-Hajnal猜想。
Characterizing Large Clique Number in Tournaments
本文研究了竞赛图中大团数的性质，证明大团数可通过有限大小的集合证明，并给出了两个简单竞赛图家族中子竞赛图的存在性。
Bounded Twin-Width Tournaments are $\dchi$-Bounded
该研究证明有界双宽锦标赛是dχ-有界的，通过提出混合参数传递定理，结合有序图相关定理与障碍，解决了特定学者的猜想。
Finding forest-orderings of tournaments is NP-complete
研究判定竞赛图是否有\(\mathcal{C}\)-反馈弧集（\(\mathcal{C}\)为所有森林的类）这一问题，证明其为NP完全问题，该研究受\(\mathcal{C}\)-反馈弧集与竞赛图结构参数联系的推动。
Subdivisions in dicritical digraphs with large order or digirth
该研究针对双临界有向图的细分包含问题，给出大阶数或大有向围长下的充分条件，证明了三项关于圈定向、一般有向图细分及TT_3细分的结论，拓展了有向图二色数相关理论。
We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichroma…
Decomposing tournaments into comparability graphs
引入有向图的偏序分解数pod，证明dic(D) ≤ diomega(D)^{pod(D)}，并应用于锦标赛，得到多项式dic有界性，解决相关问题。
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