S. Kh. Darbinyan
Let $D$ be an directed graph on $p\geq 10$ vertices with minimum degree at least $p-1$ and minimum semi-degree at least $ p/2 -1$. We present a detailed proof of the following result [13]: The digraph $D$ is pancyclic, unless some extremal cases (which are characterized).
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A near-linear Chvátal--Erdős condition for Hamilton cycles in digraphs
针对 Jackson 和 Ordaz 关于有向图哈密顿圈的猜想，本文证明了近线性上界 $f_2(a)=O(a(\log a)^4/(\log\log a)^2)$，并否证了其泛圈性猜想。
We show that for each \ell\geq 4 every sufficiently large oriented graph G with δ^+(G), δ^-(G) \geq \lfloor |G|/3 \rfloor +1 contains an \ell-cycle. This is best possible for all those \ell\geq 4 which are not divisible by 3. Surprisingly, for some other values of \ell, an \ell-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an \ell-cycle (with \ell \geq 4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity and consider \ell-cycles in general digraphs.
Chordal graphs are important in the structural and algorithmic graph theory. A digraph analogue of chordal graphs was introduced by Haskin and Rose in 1973 but has not been a subject of active studies until recently when a characterization of semicomplete chordal digraphs in terms of forbidden subdigraphs was found by Meister and Telle. Locally semicomplete digraphs, quasi-transitive digraphs, and extended semi-complete digraphs are amongst the most popular generalizations of semicomplete digraphs. We extend the forbidden subdigraph characterization of semicomplete chordal digraphs to locally …
Semicomplete multipartite weakly distance-regular digraphs
本文引入双正则队际半完全多部有向图，证明其分为三类，并刻画了所有半完全多部交换弱距离正则有向图。
Locally semicomplete weakly distance-regular digraphs
本文针对局部半完全弱距离正则有向图展开研究，在交换性假设下完成了对这类有向图的全部刻画，为弱距离正则有向图的分类研究提供了相关结果。
The 3-dicritical semi-complete digraphs
本文证明了3-双临界半完全有向图仅有有限个，并通过计算机辅助完整刻画得到全部八个此类图，其中两个为竞赛图，同时给出了3-双临界有向图最大弧数的一般上界。
Let $T$ be a digraph with vertices $u_1, \dots, u_t$ ($t\ge 2$) and let $H_1, \dots, H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1, \dots, H_t]$ is a digraph with vertex set $\{u_{i,j_i}\colon\, 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\colon\, u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ The composition $Q=T[H_1, \dots, H_t]$ is a semicomplete composition if $T$ is semicomplete, i.e. there is at least one arc between every pair of vertices. Digraph compositions generalize s…
Comparability graphs are a popular class of graphs. We introduce as the digraph analogue of comparability graphs the class of comparability digraphs. We show that many concepts such as implication classes and the knotting graph for a comparability graph can be naturally extended to a comparability digraph. We give a characterization of comparability digraphs in terms of their knotting graphs. Semicomplete comparability digraphs are a prototype of comparability digraphs. One instrumental technique for analyzing the structure of comparability graphs is the Triangle Lemma for graphs. We generaliz…
Reed showed that, if two graphs are $P_4$-isomorphic, then either both are perfect or none of them is. In this note we will derive an analogous result for perfect digraphs.
Let $k$ be an integer with $k\geq 2$. A digraph $D$ is $k$-quasi-transitive, if for any path $x_0x_1\ldots x_k$ of length $k$, $x_0$ and $x_k$ are adjacent. Suppose that there exists a path of length at least $k+2$ in $D$. Let $P$ be a shortest path of length $k+2$ in $D$. Wang and Zhang [Hamiltonian paths in $k$-quasi-transitive digraphs, Discrete Mathematics, 339(8) (2016) 2094--2099] proved that if $k$ is even and $k\ge 4$, then $D[V(P)]$ and $D[V(D)\setminus V(P)]$ are both semicomplete digraphs. In this paper, we shall prove that if $k$ is odd and $k\ge 5$, then $D[V(P)]$ is either a semi…
Let $D$ be a digraph on $p\geq 5$ vertices with minimum degree at least $p-1$ and with minimum semi-degree at least $p/2-1$. For $D$ (unless some extremal cases) we present a detailed proof of the following results [12]: (i) $D$ contains cycles of length 3, 4 and $p-1$; (ii) if $p=2n$, then $D$ is hamiltonian.
Let $D$ be a strong digraph on $n=2m+1\geq 5$ vertices. In this paper we show that if $D$ contains a cycle of length $n-1$, then $D$ has also a cycle which contains all vertices with in-degree and out-degree at least $m$ (unless some extremal cases).
The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis from 1990. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every ordered sequence of k distinct vertices of D there is a directed cycle which encounters these vertices in this order.
An out-branching $B^+_u$ (in-branching $B^-_u$) in a digraph $D$ is a connected spanning subdigraph of $D$ in which every vertex except the vertex $u$, called the root, has in-degree (out-degree) one. A {\bf good $\mathbf{(u,v)}$-pair} in $D$ is a pair of branchings $B^+_u,B^-_v$ which have no arc in common. Thomassen proved that is NP-complete to decide if a digraph has any good pair. A digraph is {\bf semicomplete} if it has no pair of non adjacent vertices. A {\bf semicomplete composition} is any digraph $D$ which is obtained from a semicomplete digraph $S$ by substituting an arbitrary digr…
Let $D$ be a digraph. We define the minimum semi-degree of $D$ as $δ^{0}(D) := \min \{δ^{+}(D), δ^{-}(D)\}$. Let $k$ be a positive integer, and let $S = \{s\}$ and $T = \{t_{1}, \dots ,t_{k}\}$ be any two disjoint subsets of $V(D)$. A set of $k$ internally disjoint paths joining source set $S$ and sink set $T$ that cover all vertices $D$ are called a one-to-many $k$-disjoint directed path cover ($k$-DDPC for short) of $D$. A digraph $D$ is semicomplete if for every pair $x,y$ of vertices of it, there is at least one arc between $x$ and $y$. In this paper, we prove that every semicomplete digra…
Paths with Prescribed Endpoints in Semicomplete and Locally Semicomplete Digraphs
本文解决半完全有向图中端点分属不同强分量的最长(x,y)-路径问题，确定局部半完全有向图连通非强情形的可能端点，给出多项式时间精确算法并分析未解决有向图的性质。
Algebraic degrees of quasi-abelian semi-Cayley digraphs
本文利用群的不可约特征标，确定了任意有限群上拟阿贝尔半凯莱有向图的分裂域和代数次数，推广了此前关于凯莱图与半凯莱有向图代数次数的相关研究。
Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, Sárközy and Selkow.
Chordal graphs are important in algorithmic graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. Unlike chordal graphs, chordal digraphs lack many structural properties such as forbidden subdigraph or representation characterizations. In this paper we introduce the notion of semi-strict chordal digraphs which form a class strictly between chordal digraphs and chordal graphs. Semi-strict chordal digraphs have rich structural properties. We characterize semi-strict chordal digraphs in terms of knotting graphs, a notion analog…
Minors of plane digraphs
本文研究了平面有向图的 semi-strong minor 性质，定义了基于分支组合的宽度度量，证明不含 H 为 semi-strong minor 的平面有向图宽度有界，而一般平面有向图宽度无界。
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…
Subgraphs with a positive minimum semidegree in digraphs with large outdegree
该研究证明了在具有大出度的有向图中存在一个子图，其最小半度数满足特定下界，并展示了该界在特定条件下是渐近最优的。
Generalized paths and cycles in semicomplete multipartite digraphs
研究半完全多部有向图中广义路径与圈的结构和算法性质，识别出可多项式时间求解或近似求解的0-1加权有向TSP实例类，同时证明两个相关圈问题为NP完全。
Let $D=(V,A)$ be a digraph and consider an arc subset $A'\subseteq A$ and an exhaustive mapping $ϕ: A\to A'$ such that $(i)$ the set of heads of $A'$ is $H(A')=V$; $(ii)$ the map fixes the elements of $A'$, that is, $ϕ|A'=Id$, and for every vertex $j\in V$, $ϕ(ω^-(j))\subset ω^-(j)\cap A'$. Then, {\it the partial line digraph} of $D$, denoted by $\mathcal{L}_{(A',ϕ)}D $ (for short $\mathcal{L}D$ if the pair $(A', ϕ)$ is clear from the context), is the digraph with vertex set $V (\mathcal{L}D)=A'$ and set of arcs $A(\mathcal{L}D) = \{(ij, ϕ(j,k)) : (j,k)\in A\}.$ In this paper we prove the foll…
For a digraph $D$, let $δ^{0}(D) = \min \{δ^{+}(D), δ^{-}(D)\}$ be the minimum semi-degree of $D$. A set of $k$ vertex-disjoint paths, $\{P_{1}, \dots, P_{k}\}$, joining a disjoint source set $S = \{s_{1}, \dots, s_{k}\}$ and sink set $T = \{t_{1}, \dots, t_{k}\}$ is called an unpaired many-to-many $k$-disjoint directed path cover ($k$-DDPC for short) of $D$, if each $P_{j}$ joins $s_{j}$ and $t_{σ(j)}$ for some permutation $σ$ on $\{1, \dots , k\}$ and $\bigcup^{k}_{j=1} V(P_{j}) = V(D)$. In this paper, we give a new proof for the following result that every digraph $D$ with $δ^{0}(D) \geq \l…
We prove that the strong immersion order is a well-quasi-ordering on the class of semi-complete digraphs, thereby strengthening a result of Chudnovsky and Seymour that this holds for the class of tournaments.
Grundy functions have found many applications in a wide variety of games, in solving relevant problems in Game Theory. Many authors have been working on this topic for over many years. Since the existence of a Grundy function on a digraph implies that it must have a kernel, the problem of deciding if a digraph has a Grundy function is NP-complete, and how to calculate one is not clearly answered. In this paper, we introduce the concept: Semi-Grundy function, which arises naturally from the connection between kernel and semi-kernel and the connection between kernel and Grundy function. We explo…
Give a digraph $D=(V(D),A(D))$, let $\partial^+_D(v)=\{vw|w\in N^+_D(v)\}$ and $\partial^-_D(v)=\{uv|u\in N^-_D(v)\}$ be semi-cuts of $v$. A mapping $φ:A(D)\rightarrow [k]$ is called a weak-odd $k$-edge coloring of $D$ if it satisfies the condition: for each $v\in V(D)$, there is at least one color with an odd number of occurrences on each non-empty semi-cut of $v$. We call the minimum integer $k$ the weak-odd chromatic index of $D$. When limit to 2 colors, use $def(D)$ to denote the defect of $D$, the minimum number of vertices in $D$ at which the above condition is not satisfied. In this pap…
The paper is concerned with directed versions of Posa's theorem and Chvatal's theorem on Hamilton cycles in graphs. We show that for each a>0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq >... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In fact, we can weaken these assumptions to (i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^- \geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G is Hamiltonian. This provides an approximate version of a …
Algebraic degrees of $n$-Cayley digraphs over abelian groups
本文研究阿贝尔群上 n-Cayley 有向图，确定其分裂域并给出代数次数的界，进而将结果推广到非阿贝尔群上的 Cayley 有向图。
Counting oriented trees in digraphs with large minimum semidegree
本文研究在有向图 $G$ 中包含多少个有向树 $T$ 的副本，证明了在最小半度条件下，$G$ 至少包含最优数量的 $T$ 的副本，并推广了无向图的相关结论。
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.
On the $k$-linkage problem for generalizations of semicomplete digraphs
本文研究半完全与拟传递等广义有向图的 $k$-链接问题，证明 $3k$-强、最小出度至少 $23k$ 的相关图类及高连通 $l$-拟传递有向图可实现 $k$-链接，并利用最大入度顶点的丰富短路径结构作为关键工具。
Oriented Trees in Digraphs without Oriented $4$-cycles
本文证明了在最小半度至少为k/2且不含4-圈的有向图中，只要最大出度和入度至少为k，就能包含任意具有k条弧的有向树，并对反有向树和树状结构给出更强结论。
On oriented cycles in randomly perturbed digraphs
该研究推广了Bohman等人关于随机扰动有向图中哈密顿圈的结果，证明在最小半度条件下，渐近几乎必然同时存在所有长度和所有定向的圈，并可将条件放宽至最小总度情形，方法基于Montgomery吸收方法的变体。
Oriented trees and paths in digraphs
本文综述了有向图中包含所有给定长度定向路径或给定大小定向树的条件，涵盖竞赛图及色数、边密度、最小出度和最小半度等条件，并指出开放问题。
Degree conditions for disjoint path covers in digraphs
本文研究多对多、一对多和一对一 $k$-DDPC 的度条件，给出若干最小半度及 Ore 型充分条件，并证明相应界的紧性或至多相差 1。
The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. After collecting together previous partial results, we refine and prove Rhodes's conjecture on the structure of the maximal groups in the flow semigroup for finite, antisymmetric, strongly connected digraphs. Building on this result, we investigate and fully describe the structure and actions of the maximal subgroups of the flow semigroup acting on all but $k$ points for all finite digraphs and graphs for all $k\geq 1$. A linear algorithm (in the number of edges) is presented to det…
In 2018, Bai, Fujita and Zhang (\emph{Discrete Math.} 2018, 341(6): 1523-1533) introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph $D$, which is a subset $S$ of vertices of $D$ such that ($a$) there exists no rainbow path for any pair of distinct vertices of $S$, and ($b$) every vertex outside $S$ can reach $S$ by a rainbow path in $D$. They showed that it is NP-hard to recognize wether an arc-coloured digraph has a RP-kernel and it is NP-complete to decided wether an arc-coloured tournament has a RP-kernel. In this paper, we give the sufficien…
Spanning subdivisions in dense digraphs
本文证明最小半度至少为 (1/2+ε)n 的 n 顶点有向图在 n≥Cm 时包含所有无孤立顶点的 m 弧有向图的细分，结果最优并强化解决了 Pavez-Signé 的猜想。
An FPT algorithm for cycle rank on semi-complete digraphs
针对半完全有向图和有限有向团宽的有向图，提出了参数化算法，在O(9^{(w+1)4^{w+2}}·n^2)时间内判定循环秩是否至多为w，并给出了半完全有向图最小反馈弧集问题的n^{O(w)}时间算法。
A container theorem for general digraphs with forbidden subdigraphs
本文提出了一般有向图在自然稀疏条件下禁止子图的容器定理，解决了传统方法无法处理的双三角形等例外情况，并扩展了之前关于有向图避免固定子图的研究。
Mader [J. Graph Theory 65 (2010) 61-69] conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $δ(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$ isomorphic to $T$ such that $G-V(T')$ is $k$-connected. The conjecture has been verified for paths, trees when $k=1$, and stars or double-stars when $k=2$. In this paper we verify the conjecture for two classes of trees when $k=2$. For digraphs, Mader [J. Graph Theory 69 (2012) 324-329] conjectured that every $k$-connected digraph $D$ with minimum semi-degree…
Polynomial Algorithms for Minimum Degree Partitions in Semicomplete Digraphs
研究半完全有向图的度约束二划分问题，通过小度证书、最小核、闭包和保护集论证以及确定性通用染色与单调重染等方法，解决了相关猜想，给出固定度阈值下的多项式时间算法。
Cycle tilings and $H$-factors in directed graphs
该研究提出了有向图中循环覆盖和H-因子的最小度数条件，并证明了关于传递竞赛因子的渐近精确Ore型结果。
On isomorphisms of $m$-Cayley digraphs
本文研究m-Cayley有向图的同构问题，通过刻画自同构群中G的正规化子并推广CI性质，定义了mCI、mPCI、mDCI和mPDCI等概念，给出Babai型判据，并分类了m≥2时的有限mDCI群和m≥4时的有限mPDCI群。
Oriented trees in digraphs without short non-directed cycles
本文将 Jiang 关于大围长图包含给定树的经典结果推广到有向图，在最小半度满足 $δ^0(D)\ge \max\{k/\ell,Δ(T)\}$ 的条件下证明大围长有向图包含所有 $k$ 边有向树，肯定回答了 Stein 和 Trujillo-Negrete 的问题。
Sparse spanning $k$-strong oriented subdigraphs in split digraphs
本文针对分裂有向图类，证明了满足最小半度条件的k-强分裂有向图存在稀疏生成k-强定向子图，改进了k-强竞赛图的相关界，同时研究了Jackson与Thomassen的猜想及MSSS$_k$问题。
It is proved that for any finite connected graph $G$, there exists an orientation of $G$ such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of $G$ (with equality if and only if $G$ is a tree). This resolves a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs.
Randomly perturbed digraphs also have bounded-degree spanning trees
本研究针对随机扰动有向图的生成树存在性问题，证明在最小半度为αn的n顶点稠密有向图中添加cn条随机边后，将以高概率包含最大度为Δ的固定定向有界度生成树，推广了无向图的相关结果并解答了已有公开问题。
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。
Many of the tools developed for the theory of tree-decompositions of graphs do not work for directed graphs. In this paper we show that some of the most basic tools do work in the case where the model digraph is a directed path. Using these tools we define a notion of a directed blockage in a digraph and prove a min-max theorem for directed path-width analogous to the result of Bienstock, Roberston, Seymour and Thomas for blockages in graphs. Furthermore, we show that every digraph with directed path width $\geq k$ contains each arboresence of order $\leq k + 1$ as a butterfly minor. Finally w…
Let $D$ be a strong digraph on $n\geq 4$ vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (*) $d(x)+d(y)\geq 2n-1$ and $min \{d(x), d(y)\}\geq n-1$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or (**) $min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or a common out-neighbour, then $D$ is hamiltonian. In this paper we show that: (i) if $D$ satisfies the condition (*) and the minimum semi-degree of $D$ at least two or (i…
A particular case of Caccetta-Häggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order $n$ with minimum out-degree at least $0.3465n$ contains a directed cycle of length at most 3 (which currently is the best result). A weaker particular case says that a digraph of order $n$ with minimum semi-degree at least $1/3n$ contains a directed triangle. In a recent paper, by using the result of Kral et al, the author proved that for $β\geq 0.343545$, any …
S. -R. Kim and F. S. Roberts (2002) introduced the following conditions $C(p)$ and $C'(p)$ for digraphs as generalizations of the condition for digraphs to be semiorders. The condition $C(p)$ (resp. $C'(p)$) is: For any set $S$ of $p$ vertices in $D$, there exists $x \in S$ such that $N^+_D(x) \subseteq N^+_D(y)$ (resp. $N^-_D(x) \subseteq N^-_D(y)$) for all $y \in S$, where $N^+_D(x)$ (resp. $N^-_D(x)$) is the set of out-neighbors (resp. in-neighbors) of $x$ in $D$. The competition graph of a digraph $D$ is the (simple undirected) graph which has the same vertex set as $D$ and has an edge bet…
Given a simple digraph $D$ on $n$ vertices (with $n\ge2$), there is a natural construction of a semigroup $\langle D\rangle$ associated with $D$. For any edge $(a,b)$ of $D$, let $a\to b$ be the idempotent of defect $1$ mapping $a$ to $b$ and fixing all vertices other than $a$; then define $\langle D\rangle$ to be the semigroup $\langle a\to b:(a,b)\in E(D)\rangle$. For $α\in \langle D \rangle$, let $\ell(D,α)$ be the minimal length of a word in $E(D)$ expressing $α$. When $D=K_n$ is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate $\ell(K_n,α)$,…
In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are precisely the digraphs that admit certain polymorphisms of interest in the study of constraint satisfaction problems; on the other hand, they are a very broad generalization of interval graphs. Bi-arc digraphs is the class of digraphs that admit conservative semilattice polymorphisms. There is much interest in understanding structures that admit particular types of polymorphisms, and especially in their recognition algorithms. (Such problems are referred to as met…
Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for every $P \in \mathcal{P}$. We say that $D$ satisfies the $α$-property if, for every maximum stable set $S$ of $D$, there exists an $S$-path partition of $D$, and we say that $D$ is $α$-diperfect if every induced subdigraph of $D$ satisfies the $α$-property. A digraph $C$ is an anti-directed odd cycle if (i) the underlying graph of $C$ is a cycle $x_1x_2 \cdo…
Sufficient conditions for a digraph to contain: a pre-Hamiltonian cycle and cycles of lengths 3 and 4
本文研究了有向图包含长度为3和4的回路以及前置哈密顿回路的充分条件，基于图的最小度数和半度数进行分析。
Generalizing well-known results of Erdős and Lovász, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $λ(H)\geq \lceil{}\frac{k-1}{k}λ(G)\rceil$, where $λ(G)$ is the edge-connectivity of $G$. In particular, together with a well-known result due to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs contains a spanning bipartite graph whose edge set decomposes into two edge-disjoint spanning trees. We show that this is best possible as it does not hold for infintely many $6$-edge-connected graphs. For directed graphs, it was shown in [6] that…
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