The average solution of a TSP instance in a graph

We define the average $k$-TSP distance $\mu_{tsp,k}$ of a graph $G$ as the average length of a shortest walk visiting $k$ vertices, i.e. the expected length of the solution for a random TSP instance with $k$ uniformly random chosen vertices. We prove relations with the average $k$-Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for $\mu_{tsp,k}(G)$ given the order of the graph.Comment: 9 pages, 3 figure

An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study

Linear-Time Algorithms for Edge-Based Problems

There is a dearth of algorithms that deal with edge-based problems in trees, specifically algorithms for edge sets that satisfy a particular parameter. The goal of this thesis is to create a methodology for designing algorithms for these edge-based problems. We will present a variant of the Wimer method [Wimer et al. 1985] [Wimer 1987] that can handle edge properties. We call this variant the Wimer edge variant. The thesis is divided into three sections, the first being a chapter devoted to defining and discussing the Wimer edge variant in depth, showing how to develop an algorithm using this variant, and an example of this process, including a run of an algorithm developed using this method. The second section involves algorithms developed using the Wimer edge variant. We will provide algorithms for a variety of edge parameters, including four different matching parameters (connected, disconnected, induced and 2-matching), three different domination parameters (edge, total edge and edge-vertex) and two covering parameters (edge cover and edge cover irredundance). Each of these algorithms are discussed in detail and run in linear time. The third section involves an attempt to characterize the Wimer edge variant. We show how the variant can be applied to three classes of graphs: weighted trees, unicyclic graphs and generalized series-parallel graphs. For each of these classes, we detail what adaptations are required (if any) and design an algorithm, including showing a run on an example graph. The fourth chapter is devoted to a discussion of what qualities a parameter has to have in order to be likely to have a solution using the Wimer edge variant. Also in this chapter we discuss classes of graphs that can utilize the Wimer edge variant. Other topics discussed in this thesis include a literature review, and a discussion of future work. There are plenty of options for future work on this topic, which hopefully this thesis can inspire. The intent of this thesis is to provide the foundation for future algorithms and other work in this area

On monophonic position sets in graphs

The general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem called the monophonic position problem, obtained by replacing ‘shortest path’ by ‘induced path’. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem