S. Pirzada, T. A. Naikoo, U. Samee, A. Ivanyi
In a directed multigraph, the imbalance of a vertex $v_{i}$ is defined as $b_{v_{i}}=d_{v_{i}}^{+}-d_{v_{i}}^{-}$, where $d_{v_{i}}^{+}$ and $d_{v_{i}}^{-}$ denote the outdegree and indegree respectively of $v_{i}$. We characterize imbalances in directed multigraphs and obtain lower and upper bounds on imbalances in such digraphs. Also, we show the existence of a directed multigraph with a given imbalance set.
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An enumeration is given of certain equivalence classes of directed multigraphs having 0 to 5 nodes with each node having 2 incoming and 2 outgoing arcs.
Branching Ratios of Input Trees for Directed Multigraphs
该研究定义了有限有向多重图节点输入树的分支比，证明其存在且等于上游节点诱导子图邻接矩阵的最大特征值，同时给出渐近性质等基础结论。
Eulerian Directed Multigraphs
本文针对有限连通非平凡有向多重图，证明了存在欧拉有向回路与非回路有向路径的充要条件，即顶点对的共回路性与特定顶点出入度关系同时成立。
For each subset of edges of a (directed multi-) graph, one may determine whether the edges can be represented as a trail. I prove that the fraction trail-representable subsets of edges is at most $O(\sqrt{\log m}/\sqrt{m})$, where $m$ is the number of edges, and show by example that the upper bound is nearly tight.
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…
It is well-known that the Chinese postman problem on undirected and directed graphs is polynomial-time solvable. We extend this result to edge-colored multigraphs. Our result is in sharp contrast to the Chinese postman problem on mixed graphs, i.e., graphs with directed and undirected edges, for which the problem is NP-hard.
Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree
本研究针对最大度不超过5的有向多重图，证明了其反馈弧集最小规模满足${\rm fas}(D)\le m/3$的紧上界，强化了Hanauer的猜想并改进了已有结果，同时给出了按顶点数计的紧界常数$c$的范围。
In a directed graph, the imbalance of a vertex is its outdegree minus its indegree. We characterize the sequences that are realizable as the sequence of imbalances of a simple directed graph. Moreover, a realization of a realizable sequence can be produced by a greedy algorithm.
We prove that if a directed multigraph $D$ has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of less than $2t$ arcs in $D$ which meets all directed triangles in $D$, except in the trivial case $t=0$. This answers affirmatively a question of Tuza from 1990.
We consider the strongly connected components (SCCs) of a uniform directed graph on $n$ vertices with i.i.d. in- and out-degree pairs distributed as $(D^-,D^+)$, with $\mathbb E[D^+]=\mathbb E[D^-]=μ$. We condition on equal total in- and out-degree. A phase transition for the emergence of a giant SCC is known to occur at the critical value $\mathbb E[D^-D^+] = μ$. We study the model at this critical value and, additionally, require that $\mathbb E[(D^-)^i(D^+)^j]<\infty$ for all $i+j\leq 3$, and for $(i,j)=(1,3)$ and $(i,j)=(3,1)$. We show that, under these conditions, the SCCs ranked by decre…
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. We introduce a heuristic general variable neighborhood search algorithm for this problem based on finding a vertex-disjoint cycle cover of the multigraph through reduction to perfect matching and several cycle merging operations. The algorithm has a one-sided error: the answer "not adjacent" is…
Linear Extensions of Rotor-Routing in Directed Graphs: Reachability Problems
本文提出了一种扩展旋转路由的框架，研究了不同路由模型在多图中的可达性问题，证明了GRM多图的合法可达性为NP难，而循环GRM路由具有多项式时间算法。
Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs
本文提出在加权有向无环多图上扩展莫比乌斯变换和沙普利值，以处理向量值函数，引入投影算子和新公理，实现对复杂系统结构的分解与归因。
This paper characterizes the injective and projective objects in the category of directed multigraphs, or quivers. Further, the injective envelope and projective cover of any quiver in this category is constructed.
Mader proved that every strongly $k$-connected $n$-vertex digraph contains a strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, where the equality holds for the complete bipartite digraph ${DK}_{k,n-k}$. For dense strongly $k$-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly $k$-connected $n$-vertex digraph $D$ contains a strongly $k$-connected spanning subgraph with at most $kn + 800k(k+\overlineΔ(D))$ edges, where $\overlineΔ(D)$ denotes the maximum degree of the complement of the underlying undirected gra…
In this paper, we extend the classical arithmetic defined over the set of natural numbers N, to the set of all finite directed connected multigraphs having a pair of distinct distinguished vertices. Specifically, we introduce a model F on the set of such graphs, and provide an interpretation of the language of arithmetic L={0,1,<=,+,x} inside F. The resulting model exhibits the property that the standard model on N embeds in F as a submodel, with the directed path of length n playing the role of the standard integer n. We will compare the theory of the larger structure F with classical arithme…
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。
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