A matrix for counting paths in acyclic colored digraphs
Sudip Bera
本文提出一种与k-着色无环有向图关联的矩阵，并证明其行列式可枚举图中所有路径，从而建立矩阵行列式与路径计数之间的对应关系。
在本文中，我们引入一个与k-着色无环有向图Γ_{k, R}相关联的矩阵A_{Γ_{k, R}}，使得det(A_{Γ_{k, R}})枚举该有向图Γ_{k, R}中的路径。
In this paper, we introduce a matrix $A_{Γ_{k, R}}$ associate with a $k$-colored acyclic digraph $Γ_{k, R}$ such that $\text{det}(A_{Γ_{k, R}})$ enumerates the paths in the digraph $Γ_{k, R}.$
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A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a properly colored directed path in $D$. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.
A descent of a labeled acyclic digraph is a directed edge $x\to y$ with $x>y$. In this paper, we find a recurrence for the number of labeled acyclic digraphs with a given number of descents.
An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.
A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments, strong digraphs, and acyclic digraphs by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.
On the singularity and the inverse of 3-colored digraphs
本文研究了3色有向图的非奇异性和逆矩阵特性，探讨了非奇异单环和双环有向图的特征，并分析了其逆矩阵的零对角线条件及可逆性。
Kernels in digraphs with colored vertices
该研究提出顶点着色有向图中核的推广概念——上着色核，并推导得出若干有向图族及特定有向图积具备上着色核的充要条件。
We complete the characterization of the digraphs $D$ for which the induced $D$-removal lemma has polynomial bounds, answering a question of Alon and Shapira. We also study the analogous problem for $k$-colored complete graphs. In particular, we prove a removal lemma with polynomial bounds for Gallai colorings.
Let $D=(V,A)$ be an acyclic digraph. For $x\in V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V \rightarrow [1,|V|] $ that has the property that for all $x,y\in V$ if $(x,y)\in A$, then $g(x) < g(y)$. We prove that for every acyclic ordering $g$ of $D$ the following inequality holds: \[\sum_{x\in V} e_{_{D}}(x)\cdot g(x) ~\geq~ \frac{1}{2} \sum_{x\in V}[e_{_{D}}(x)]^2~.\] The class of acyclic digraphs for which equality holds is determined as the class of comparbility digraphs of posets o…
We study $k$-colored kernels in $m$-colored digraphs. An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that (i) from every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and (ii) for every $u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper, we prove that for every integer $k\geq 2$ there exists a $% (k+1)$-colored digraph $D$ without $k$-colored kernel and if every directed cycle of an $m$-colored digraph is monochromatic, then it has a $k$-colored …
An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K $ of its vertices such that for every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and for every $% u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper we prove that an $m$-colored semicomplete $r$-partite digraph $D$ has a $k$-colored kernel provided that $r\geq 3$ and {enumerate} [(i)] $k\geq 4,$ [(ii)] $k=3$ and every $\overrightarrow{C}_{4}$ contained in $D$ is at most 2-colored and, either every $\overrightarrow{C}_{5…
We show that the number of acyclic directed graphs with n labeled vertices is equal to the number of n X n (0,1)-matrices whose eigenvalues are positive real numbers.
We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes and multiply constant-weight codes. This was achieved via an interesting application of the theory of decomposition of edge-colored digraphs.
Acyclic and complete coloring of digraphs with the minimum and maximum possible numbers of colors
本文研究有向图无环与完全着色中颜色数的极值问题，构造了任意给定二色数 r 和双色数 t（r≤t）的非对称有向图，并给出非对称情形下的二次上界。
Let $D$ be an arc-colored digraph. The arc number $a(D)$ of $D$ is defined as the number of arcs of $D$. The color number $c(D)$ of $D$ is defined as the number of colors assigned to the arcs of $D$. A rainbow triangle in $D$ is a directed triangle in which every pair of arcs have distinct colors. Let $f(D)$ be the smallest integer such that if $c(D)\geq f(D)$, then $D$ contains a rainbow triangle. In this paper we obtain $f(\overleftrightarrow{K}_{n})$ and $f(T_n)$, where $\overleftrightarrow{K}_{n}$ is a complete digraph of order $n$ and $T_n$ is a strongly connected tournament of order $n$.…
We consider vertex coloring of an acyclic digraph $\Gdag$ in such a way that two vertices which have a common ancestor in $\Gdag$ receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding {\em down-chromatic number} and derive an upper bound as a function of $D(\Gdag)$, the maximum number of descendants of a given vertex, and the degeneracy of the corresponding hypergraph. Finally we determine an asymptotically tight upper bound of the down-chromatic number in terms of the number of vertices …
Large flames in rooted acyclic digraphs without backward-infinite paths
研究了无反向无限路径的无限有向图中火的性质，证明了Szeszlér定理的扩展形式及Lovász定理的无限推广
Thomassen's theorem on the two-linkage problem in acyclic digraphs: a shorter proof
该研究针对无环有向图的两链接问题，给出了Thomassen 1985年提出的经典定理的更简短证明，该定理刻画了无环有向图中不存在两条指定不相交有向路径的充要条件。
A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most $f(d)$ monochromatic $d$-dominating subgraphs. Note that the existence of $f(d)$ is not obvious -- indeed, the question which motivated this paper was simply to determine whether $f(d)$ is bounded, even for $d=2$. We answer this question affirmatively for all $d\geq 2$, proving $4\leq f(2)\le …
I present an algorithm that, given a number $n \geq 1$, computes a compact representation of the set of all noncrossing acyclic digraphs with $n$ nodes. This compact representation can be used as the basis for a wide range of dynamic programming algorithms on these graphs. As an illustration, along with this note I am releasing the implementation of an algorithm for counting the number of noncrossing acyclic digraphs of a given size. The same tabulation can be modified to count other classes of combinatorial structures, including weakly connected noncrossing acyclic digraphs, general noncrossi…
We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $ψ\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $φ\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $φ=f\circψ$. This implies the main res…
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…
Hefner [K. A. S. Hefner, K. F. Jones, S. -R. Kim, R. J. Lundgren and F. S. Roberts: $(i,j)$ competition graphs, Discrete Applied Mathematics, 32, (1991) 241-262] characterized acyclic digraphs each vertex of which has inderee and outdegree at most two and whose competition graphs are interval. They called acyclic digraphs each vertex of which has inderee and outdegree at most two $(2,2)$ digraphs. In this paper, we study the phylogeny graphs of $(2,2)$ digraphs. Especially, we give a sufficient condition and necessary conditions for $(2,2)$ digraphs having chordal phylogeny graphs. Phylogeny g…
Acyclic sets and colorings in digraphs under restrictions on degrees and cycle lengths
本文研究了有向图中无环集和着色问题在度数和环长限制下的性质，证明了在随机有向图中无环集大小的渐进行为，并提出了相关猜想。
We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from…
The dichromatic number of a digraph $G$ is the smallest integer $χ_a(G)$ such that the vertex set of $G$ can be partitioned into $χ_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of di…
In this note we derive enumerative formulas for several types of labelled acyclic directed graphs by slight modifications of the familiar recursive formula for simple acyclic digraphs. These considerations are motivated by, and based upon, recent combinatorial results in geometric topology obtained by S.Choi, who established exact correspondences between acyclic digraphs and so-called small covers over hypercubes and related polytopes. In particular, we show that the number of equivalence classes of small covers over the cartesian product of $n$ copies of an $r$-simplex is equal to the number …
Masuda (2008) provided the characterization of real Bott manifolds in terms of three operations on upper triangular matrices. We provide a combinatorial characterization of real Bott manifolds up to diffeomorphism in terms of operations on directed acyclic graphs. Our observation leads to several new invariants of real Bott manifolds.
Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $Δ\geq 2$ and any sufficiently large $n$, there exists an acyclic digraph…
An acyclic homomorphism of a digraph $C$ to a digraph $D$ is a function $ρ\colon V(C)\to V(D)$ such that for every arc $uv$ of $C$, either $ρ(u)=ρ(v)$, or $ρ(u)ρ(v)$ is an arc of $D$ and for every vertex $v\in V(D)$, the subdigraph of $C$ induced by $ρ^{-1}(v)$ is acyclic. A digraph $D$ is a core if the only acyclic homomorphisms of $D$ to itself are automorphisms. In this paper, we prove that for certain choices of $p(n)$, random digraphs $D\in D(n,p(n))$ are asymptotically almost surely cores. For digraphs, this mirrors a result from [A. Bonato and P. Prałat, The good, the bad, and the great…
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H-$colored digraph). If $W=(x_{0},\ldots,x_{n})$ is an open walk in $D$ and $i\in \{1,\ldots,n-1\}$, we say that there is an obstruction on $x_{i}$ if $(color(x_{i-1},x_{i}),color(x_{i},x_{i+1}))\notin A(H)$. If $S\subseteq V(D)$, we say that $S$ is a $(k,l,H)$-kernel by walks if for every pair of different vertices in $S$, every walk between them has at least $k-1$ obstructions, and for every $x\in V(D)\setminus S$ there exists an $xS$-walk with at mo…
Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order $n$ for which the size of the maximum acyclic set is at most $\lceil \frac{n+1}{2} \rceil$, for any $n$. This disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.
\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The \emph{dichromatic number} $χ_d(D)$ of $D$ is the minimum number of colors needed in a coloring of $D$. Besides obtaining many results and bounds for $χ_d(D)$ analogous to that of chromatic number of a graph, we prove $χ_d(D)=1$ if $D$ is acyclic. New notions of sequential colorings of graphs/digraphs are introduced. A characterization of acyclic digraph is obtained …
Cut covers of acyclic digraphs
该研究证明无环有向图割覆盖的入度条件是紧的，给出有向路径幂图割覆盖的最大幂次范围，并证明相关同态有向图的割覆盖判定问题是NP完全的。
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops with a coloring of its arcs $c:A(D) \rightarrow V(H)$ ($D$ is said to be an $H$-colored digraph). A directed path $W$ in $D$ is said to be an $H$-path if and only if the consecutive colors encountered on $W$ form a directed walk in $H$. A subset $N$ of vertices of $D$ is said to be an $H$-kernel if (1) for every pair of different vertices in $N$ there is no $H$-path between them and (2) for every vertex $u$ in V($D$)$\setminus$$N$ there exists an $H$-path in $D$ from $u$ to $N$. Under this definition an $H$-kernel is a ke…
Acyclic subgraphs of digraphs with high chromatic number
研究了具有高色数的有向图的无环子图色数，证明了在特定条件下该色数下界为$n^{5/9-o(1)}$，并改进了现有界。
Competition-common enemy graphs of degree-bounded digraphs
本文完整刻画了⟨2,2⟩有向图的竞争-公共敌图（CCE图），证明分量数≤7的无环⟨2,2⟩有向图的CCE图为区间图且界紧，拓展了度有界有向图相关图类的研究。
Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of…
A $k$-majority digraph is a directed graph created by combining $k$ individual rankings on the same ground set to form a consensus where edges point in the direction indicated by a strict majority of the rankings. The $k$-majority digraph is used to model voting scenarios, where the vertices correspond to options ranked by $k$ voters. When $k$ is odd, the resulting digraph is always a tournament, called $k$-majority tournament. Let $f_k(n)$ be the minimum, over all $k$-majority tournaments with $n$ vertices, of the maximum order of an induced transitive sub-tournament. Recently, Milans, Schrei…
F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a) are considered. For such specific sequences a new interpretation with respect to Kwasniewski general combinatorial interpretation of F-nomial coefficients is unearhed. Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as a-multigraphs G(a,n,k). An explicit relation bet…
We investigate the complexity of generalizations of colourings (acyclic colourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), for the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.
Acyclic orientations of mixed graphs
研究无环混合图的定向完成问题，证明若干判定问题的复杂性，并给出一个多项式算法，提出开放问题。
Orthogonality between acyclic subdigraphs and paths in digraphs
本文研究了有向图中无环子图与路径的正交性，证明了在替换稳定集为诱导无环子图后，可以放松两个开放猜想。
Generalized tableaux over arbitrary digraphs and their associated differential equations
本文在任意多重有向图上推广无环序的排列与计数器，证明嵌套有向路径下计数器生成函数为多项式与指数函数之积，并借助两行排列有向图构造出具有多项式解的低阶线性微分方程，特别得到与 Catalan 数相关的 Catalan 微分方程和正交 Catalan 多项式。
A Mathon-type construction for digraphs and improved lower bounds for Ramsey numbers
本文提出有向图的 Mathon 型构造，证明其与 k 次幂 Paley 有向图的联系，并据此改进多色有向 Ramsey 数的下界。
Improved lower bounds for the maximum order of an induced acyclic subgraph
本文改进了有向图最大诱导无环顶点集大小的下界，通过结合局部度数数据和反馈顶点集方差分析，提出更精确的下界方法。
Extensions of results on phylogeny graphs of degree bounded digraphs
该研究延续贝叶斯网络证据传播相关的系统发育图研究方向，扩展了已有成果，给出弦$(i,2)$系统发育图的必要条件，并通过列出禁用诱导子图给出$(i,j)$系统发育图的必要条件。
Recent investigations in computational biology focus on a family of 2-colored digraphs, called 2-colored best match graphs, which naturally arise from rooted phylogenetic trees. Actually the defining properties of such graphs are unexpectedly unusual in graph theory, and they were established only recently after the discovery of their links to evolutionary relatedness via phylogenetic trees. In this paper several results are obtained on 2-colored best match graphs which well fit in the mainstream of graph theory.
We show that any planar drawing of a forest of three stars whose vertices are constrained to be at fixed vertex locations may require $Ω(n^\frac{2}{3})$ edges each having $Ω(n^\frac{1}{3})$ bends in the worst case. The lower bound holds even when the function that maps vertices to points is not a bijection but it is defined by a 3-coloring. In contrast, a constant number of bends per edge can be obtained for 3-colored paths and for 3-colored caterpillars whose leaves all have the same color. Such results answer to a long standing open problem.
Let $D$ be a digraph and let $λ(D)$ denote the number of vertices in a longest path of $D$. For a pair of vertex-disjoint induced subdigraphs $A$ and $B$ of $D$, we say that $(A,B)$ is a partition of $D$ if $V(A)\cup V(B)=V(D).$ The Path Partition Conjecture (PPC) states that for every digraph, $D$, and every integer $q$ with $1\leq q\leqλ(D)-1$, there exists a partition $(A,B)$ of $D$ such that $λ(A)\leq q$ and $λ(B)\leqλ(D)-q.$ Let $T$ be a digraph with vertex set $\{u_1,\dots, u_t\}$ and for every $i\in [t]$, let $H_i$ be a digraph with vertex set $\{u_{i,j_i}\colon\, j_i\in [n_i]\}$. The {…
Blow-ups and extensions of trees in tournaments
本研究将有向树的线性不可避免性推广，证明了添加至多k个通用顶点的有向树及有向树的k-膨胀图均为线性不可避免，并给出了对应的竞赛图阶数界。
Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $ρ: A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0, \ldots , n-1 \}$, we say that there is an obstruction on $x_{i}$ whenever $(ρ(x_{i-1}, x_{i}), ρ(x_{i}, x_{i+1})) \notin A(H)$ (when $x_{0} = x_{n}$ the indices are taken modulo $n$). We denote by $O_{H}(W)$ the set $\{ i \in \{0, \ldots , n-1 \} :$ there is an obstruction on $x_{i} \}$. The $H$-length of $W$, denoted by $l_{H}(W)$, is defined by $|O_{H}(W)|+1$ whe…
In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.
Minimal Path and Acyclic Models in the Path Complex
本文研究有向图路径复形的极小路径结构，分析极小3-路径实例，证明极小路径支撑子有向图具有无环路径同调，并探讨该无环模型的应用。
An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices $(u, v)$, the digraph $D$ contains a directed path (a directed walk) from $u$ to $v$ such that arcs adjacent on that path (on that walk) have distinct colors. The proper connection number $\overrightarrow{pc}(D)$ (the proper-walk connection number $\overrightarrow{wc}(D)$) of a digraph $D$ is the minimum number of colours to make $D$ properly connected (properly-walk connected). We prove that $\overrightarrow{pc}(C_n(S)) \leq 2$ for every circulant digraph $C_n(S)$ with $S\subseteq\{1,\ldots ,n-…
Classification of real Bott manifolds and acyclic digraphs
本文通过矩阵运算和图运算刻画实Bott流形的仿射微分同胚分类，证明其乘积分解唯一性及环面秩猜想成立。
In 1982 Víctor Neumann-Lara introduced the dichromatic number of a digraph $D$ as the smallest integer $k$ such that the vertices $V$ of $D$ can be colored with $k$ colors and each color class induces an acyclic digraph. Later a flow theory for the dichromatic number transferring Tutte's theory of nowhere-zero flows (NZ-flows) from classic graph colorings has been developed by Hochstättler. The purpose of this paper is to pursue this analogy by introducing a new definition of algebraic Neumann-Lara-flows (NL-flows) and a closed formula for their polynomial. Furthermore we generalize the Equiva…
The Path Partition Conjecture and 3-(quasi-)Transitive Digraphs
本文针对3-传递和3-拟传递有向图，引入半无环有向图概念刻画其结构，并将路划分猜想的成立范围推广至扩展半无环有向图类。
An equality for balanced digraphs
本文证明平衡有向图中，不含环且每个顶点到指定顶点s有路径的k边子集（称为s-收敛）的数量与s无关，推广了关于生成树形图、无环定向和最大无环子图的结果。
We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive integers; denote by $S(v)$ the sum of labels over all neighbors of each vertex $v$. The labeling $f$ is called \emph{topological additive numbering} if $S(u) < S(v)$ for each arc $(u,v)$ of the digraph. The problem asks to find the minimum number $k$ for which $D$ has a topological additive numbering with labels belonging to $\{ 1, \ldots, k \}$, denoted by $η_t…
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。
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