Homomorphism complexes, reconfiguration, and homotopy for directed graphs

The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs $G$ and $H$, we consider the polyhedral complex $\text{Hom}(G,H)$ that parametrizes the directed graph homomorphisms $f: G \rightarrow H$. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals. We study examples of directed Hom complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of $\text{Hom}(G,T_n)$ for $T_n$ a tournament. Finally, we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified statements and proofs, other corrections and minor revisions incorporating comments from referee

The Conditional Connectivity of Graphs

图的连通性是图论的一个基本研究课题，它与网络的可靠性（容错性）密切相关.图的经典的连通性是用连通度和 边连通度来度量的.图的连通度（边连通度）就是使图不连通所要删除的最小的点（边）数，显然它反映了相应的网络的容错性. 在通信网络的可靠性的进一步研究中，人们需要了解两个具有相同的连通度或边连通度的图，哪个可靠性更高？ 为了深入研究网络的可靠性或容错性，人们提出了许多条件连通性的概念. Harary在1983年引入条件连通度的概念\cite{2}.设图$G=(V，E)$，$P$是图的某种性质，S\subseteqV(G).$G$的条件连通度 $\kappa(G:P)$=min...The connectivity of graph is an element topic in the research of graph theory, it is closely related to the network reliability (fault tolerance). The classical connectivity concept of graph is connectivity and edge connectivity. Connectivity (edge connectivity) is the minimum number of vertices (edges) whose deletion makes the graph disconnected. Obviously, they reflect the corresponding net...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：1912008015045

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Fuelling the zero-emissions road freight of the future: routing of mobile fuellers

The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios

Connectivity of the Mycielskian of a digraph

NSFC [10831001]In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph mu (G), which is called the Mycielskian of G. This work investigates the vertex connectivity and arc connectivity of the Mycielskian of a digraph D. This generalizes the recent results due to Balakrishnan and Raj [R. Balakrishnan, S.F. Raj, Connectivity of the Mycielskian of a graph, Discrete Math, 308 (2008). 2607-2610]. (C) 2009 Elsevier Ltd. All rights reserved