Janusz Adamus
We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices.
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$3$-diregular circulant digraph on $12$ vertices is a counterexample of Jackson's conjecture about hamiltonicity of diregular digraphs
We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For k greater than or equal to 2, a bipartite digraph D with colour classes of cardinalities k is hamiltonian if the sum of degrees of vertices u and v is at least 3k+1 for every pair of vertices u, v such that D does not contain the arc uv nor vu. As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a balanced bipartite digraph D on 2k vertices is hamiltonian if its minimal degree is at least (3k + 1)/2.
We prove that a strongly connected balanced bipartite digraph $D$ of order $2a$ is hamiltonian, provided $a\geq3$ and $d(x)+d(y)\geq 3a$ for every pair of vertices $x$, $y$ with a common in-neighbour or a common out-neighbour in $D$.
The symmetric group is generated by σ = (1 2 ... n) and τ = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n.
R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the following problem. \textbf{Problem.} Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 8$. Suppose that $d(x)\geq 2a-k$, $ d(y)\geq a+k$ or $d(y)\geq 2a-k$, $ d(x)\geq a+k$ for every pair of vertices $x,y$ with a common out-neighbour, where $2 \leq k\leq a/2$. Is $D$ Hamiltonian? In this paper, we prove that if a digraph $D$ satisfies the conditions of this problem, then (i) $D$ contains a cycle factor, (ii) for every vertex $x\in V(D)$ there exists a vertex $y\in V(D…
Non-Hamiltonian 2-regular Digraphs -- Residues
本文研究非哈密顿2-正则有向图的性质，通过引入双共轭、排除集和余数的概念，证明了非哈密顿性的必要且充分条件。
Rainbow Hamiltonicity in uniformly coloured perturbed digraphs
本文证明了在边均匀着色的随机扰动有向图中，当底图最小半度满足条件时，以高概率存在彩虹有向哈密顿圈，改进了此前无向图设置下的相关结论。
In his seminal paper from 1952 Dirac showed that the complete graph on $n\geq 3$ vertices remains Hamiltonian even if we allow an adversary to remove $\lfloor n/2\rfloor$ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on $n\geq 3$ vertices with minimum in- and out-degree at least $n/2$ contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (…
A generalized de Bruijn digraph generalizes a de Bruijn digraph to the case where the number of vertices need not be a pure power of an integer. Hamiltonian cycles in these digraphs thus generalize regular de~Bruijn cycles, and we will thus refer to them simply as de Bruijn cycles. We define the cross-join to be the graph with all de Bruijn cycles as vertices, there is an edge between two of these vertices if one can be obtained from the other via a cross-join operation. We show that the cross-join graph is connected. This in particular means that any regular de Bruijn cycle can be cross-joine…
At least two different notions have been published under the name "majority domination in graphs": Majority dominating functions and majority dominating sets. In this work we extend the former concept to digraphs. Given a digraph $D=(V,A),$ a function $f : V \rightarrow \{-1,1\}$ such that $f(N^+[v])\geq1$ for at least half of the vertices $v$ in $V$ is a majority out-dominating function (MODF) of $D.$ The weight of a MODF $f$ is $w(f)=\sum\limits_{v\in V}f(v),$ and the minimum weight of a MODF in $D$ is the majority out-domination number of $D,$ denoted $γ^+_{maj}(D).$ In this work we introdu…
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 …
The concept of majority domination in graphs has been defined in at least two different ways: As a function and as a set. In this work we extend the latter concept to digraphs, while the former was extended in another paper. Given a digraph $D=(V,A),$ a set $S\subseteq V$ is a \textit{majority out-dominating set} (MODS) of $D$ if $|N^+[S]|\geq \frac {n}{2}.$ The minimum cardinality of a MODS in $D$ is the {\it set majority out-domination number} $γ^+_{m}(D)$ of $D.$ In this work we introduce these concepts and prove some results about them, among which the characterization of minimal MODSs.
We prove a sharp Ore-type criterion for hamiltonicity of balanced bipartite digraphs: A bipartite digraph D, with colour classes of cardinality N, is hamiltonian if, for every pair of vertices u and v from opposite colour classes of D such that the arc uv is not in D, the sum of the outdegree of u and the indegree of v is greater than or equal to N+2.
Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ the sets of in-neighbors of $x$ and $y$ are disjoint. The set $S$ is a dominating set (an open dominating set) in $D$ if every vertex not in $S$ (in $V$) has an in-neighbor in $S$. Moreover, a dominating set $S$ is called a total dominating set if the subgraph induced by $S$ has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We pr…
Given a graph G = (V,E), a subset S of V is dominating if for every v in V - S there exists u in S such that uv is in E. A dominating subset S of V is secure if for every v in V - S there exists u in S such that (S - {u}) U {v} is dominating. In this work we extend the concept of secure dominating set to digraphs in four different ways, all of them with interesting applications, and prove some results regarding each of them.
A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts which can be considered as two extensions of double domination in graphs to digraphs, along wit…
We study the eternal dominating number and the m-eternal dominating number on digraphs. We generalize known results on graphs to digraphs. We also consider the problem "oriented (m-)eternal domination", consisting in finding an orientation of a graph that minimizes its eternal dominating number. We prove that computing the oriented eternal dominating number is NP-hard and characterize the graphs for which the oriented m-eternal dominating number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs and different kinds of…
Let $1\le n\in\Z$. {\it Worst-case efficient dominating sets in digraphs} are conceived so that their presence in certain strong digraphs $\vec{ST}_n$ corresponds to that of efficient dominating sets in star graphs $ST_n$: The fact that the star graphs $ST_n$ form a so-called dense segmental neighborly E-chain is reflected in a corresponding fact for the digraphs $\vec{ST}_n$. Related chains of graphs and open problems are presented as well.
A dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of a dominating (respectively, total dominating) set of $D$ is the domination (respectively, total domination) number of $D$, denoted $γ(D)$ (respectively,$γ_t(D)$). The maximum number of pairwise disjoint closed (respectively,open) in-neighborhoods of $D$ is denoted by $ρ(D)$ (respectively,$ρ^{\rm o}(D)$). We prove that in digraphs whose underlying graphs hav…
For a positive integer $k$, a $k$-rainbow dominating function ($k$RDF) on a digraph $D$ is a function $f$ from the vertex set $V(D)$ to the set of all subsets of $\{1,2,\ldots,k\}$ such that for any vertex $v$ with $f(v)=\emptyset$, $\bigcup_{u\in N^-(v)}f(u)=\{1,2,\ldots,k\}$, where $N^-(v)$ is the set of in-neighbors of $v$. The weight of a $k$RDF $f$ is defined as $\sum_{v\in V(D)}|f(v)|$. A $k$RDF $f$ on $D$ with no isolated vertex is called a total $k$-rainbow dominating function if the subdigraph of $D$ induced by the set $\{v\in V(D):f(v)\ne\emptyset\}$ has no isolated vertex. The total…
Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 4$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. In this paper we prove: (i). If $a\geq 4$ and $ max\{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, $D$ then contains a cycle of length $2a-2$ or $D$ is a directed cycle. (ii). If $D$ contains a cycle of length $2a-2\geq 6$ and $max \{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then for any $…
We prove two sharp sufficient conditions for hamiltonian cycles in balanced bipartite directed graph. Let $D$ be a strongly connected balanced bipartite directed graph of order $2a$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. (i) {\it If $a\geq 4$ and $max \{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, then either $D$ is hamiltonian or $D$ is isomorphic to one exceptional digraph of order eight.} (ii) {\it If $a\geq 5$ and $d(x)+d(y)\geq 4a-3$ f…
Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 10$ other than a directed cycle. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. In this paper we prove: If $ max\{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then $D$ contains cycles of all lengths $2,4, \ldots , 2a-2$ or $D$ is isomorphic to a certain digraph of order ten which we specify.
Hamilton Cycles in Dense Regular Digraphs and Oriented Graphs
本文证明了度数至少为 $(1/4 + \varepsilon)n$ 的稠密正则定向图包含哈密顿圈，确立了 Jackson 1981 年猜想的近似版本，并拓展了正则有向图哈密顿性的相关结果。
Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 10$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. In this paper we prove: {\it If the underlying undirected graph of $D$ is not 2-connected and $max\{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then $D$ contains a cycle of length $2a-2$ unless $D$ is isomorphic to a certain digraph of order ten which we specify.
A dominating set $D$ in a digraph is a set of vertices such that every vertex is either in $D$ or has an in-neighbour in $D$. A dominating set $D$ of a digraph is locating-dominating if every vertex not in $D$ has a unique set of in-neighbours within $D$. The location-domination number $γ_L(G)$ of a digraph $G$ is the smallest size of a locating-dominating set of $G$. We investigate upper bounds on $γ_L(G)$ in terms of the order of $G$. We characterize those digraphs with location-domination number equal to the order or the order minus one. Such digraphs always have many twins: vertices with t…
Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A_1,...,A_t of independent vertices. A set $U=\cup_{i\in S} A_i$ is called a dominating set of size |S| if for any vertex $v\in \cup_{i\notin S} A_i$ there is a w in U such that (w,v) is in E(D). Let beta(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h=h(beta(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colo…
Cycles of lengths 3 and n-1 in digraphs under a Bang-Jensen-Gutin-Li type conditon
本文研究有向图哈密顿性的Bang-Jensen-Gutin-Li型条件，建立了一个不仅蕴含哈密顿圈还蕴含3-圈和(n-1)-圈的条件，并猜想其蕴含所有长度圈的存在。
Perfect codes in 2-valent Cayley digraphs on abelian groups
研究阿贝尔群上强连通2价Cayley有向图的完美码问题，完成图的分类并完全确定其所有完美码。
Extremal digraphs for open neighbourhood location-domination and identifying codes
本文研究开邻域定位支配集等于全部顶点的极值有向图，证明其一般结构性质并给出构造方法，同时为已有结果提供新证明并刻画极值有向树。
A sufficient condition for a balanced bipartite digraph to be hamiltonian
本文为平衡二分有向图提出新的充分条件：若每个支配对满足特定的度下界，则强平衡二分有向图必含哈密顿圈，且下界最优。
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。
A new condition on dominated pair degree sum for a digraph to be supereulerian
本文在已有强有向图超欧拉性的不相邻顶点对度数和条件基础上，进一步针对受控与支配型不相邻顶点对研究对应的度数和条件，拓展超欧拉图的判定依据。
An \textit{Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. The \textit{weight} of an Italian dominating function $f$ is the value $ω(f) = f(V(D)) = \sum_{u \in V(D)} f(u)$. The \textit{Italian domination number} of a digraph $D$, denoted by $γ_I(D)$, is the minimum taken over the weights of all Italian dominating functions on $D$. The \textit{Italian bondage number} of a …
Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph is a dominating subset $C$ of vertices such that all distinct subsets of vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood within $C$. In this paper, we prove that every $k$-iterated line digraph of minimum in-degree at least 2 and $k\geq2$, or minimum in-degree at least 3 and $k\geq1$, admits a $(1,\le \ell)$-identifying code with $\ell\leq2$, and in any case it does not admit a $(1,\le \ell)$-identifying code for $\ell\geq3$. Moreover, we find that the identifying number of a line digraph …
An end degree for digraphs
本文为无限有向图末端定义了度数并扩展出兼顾支配顶点的组合末端度数，通过末端穷尽序列刻画其特征，解决了Zuther的问题。
Greedy and randomized heuristics for optimization of k-domination models in digraphs and road networks
本文将道路加油设施选址建模为可达性有向图中的 k-支配集问题，提出贪婪与随机化启发式算法并证明新上界，在英国西米德兰兹郡道路网络上验证了其高效性。
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。
A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are characterized uniquely by the in-neighbours they have in $S$, i.e. for every two vertices $u$ and $v$ that are not in $S$, there exists a vertex $s\in S$ that dominates exactly one of them. The size of a smallest set of a directed graph $D$ which is both locating and dominating is denoted by $γ^{LD}(D)$. Foucaud, Heydarshahi and Parreau proved that any twin…
Strong chordality in tournaments and multipartite tournaments with possible loops
本文研究可能带环的多部锦标赛的强弦性，给出多项式时间识别算法与禁用导出子图刻画，并证明强弦有向图中全控制集最小规模等于不相交入邻域最大数目且可在线性时间内计算。
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, trees, DAGs, cycles, and bipartite graphs. We also provide the idomatic number for special cases of some of these families of digraphs.
An integer distance digraph is the Cayley graph $Γ(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $Γ(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $Γ(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$…
We define strongly chordal digraphs, which generalize strongly chordal graphs and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0,1 matrices that admit a simultaneous row and column permutation avoiding the Γ matrix. In general, it is not clear if these digraphs can be recognized in polynomial time, and we focus on symmetric digraphs (i.e., graphs with possible loops), tournaments with possible loops, and balanced digraphs. In each of these cases we give a polynomial-time recognition algorithm and a forbidden induced subgraph characteriz…
Local Fan-type obstructions and Ore-type conditions for rainbow Hamiltonicity in graph systems
该研究针对图系统，提出局部Fan型与Ore型条件，证明满足对应条件可保证存在彩虹哈密顿圈，Fan型假设还可得到彩虹哈密顿路径和彩虹$(n-1)$-圈，对角系统下条件退化为经典条件。
On the total Italian domination number in digraphs
本文研究有向图的全意大利支配数，给出其上下界，并确定笛卡尔积 $P_2\Box P_n$ 和 $P_3\Box P_n$ 的精确全意大利支配数。
An Italian dominating function on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. In this paper, we determine the exact values of the Italian domination numbers of some products of directed cycles.
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