Research Assistant, The Threshold Dimension and Threshold Strong Dimension of Cartesian Products of Graphs
About the role
Research Assistants are normally students, in a University of Winnipeg undergraduate program, employed to assist with tasks related to research. Work is performed within defined guidelines and under supervision of a more senior researcher or faculty member.
Position Overview:
The goal of this summer research project is to solve an open problem in graph theory using algebraic tools. The problem is that of determining the threshold dimension and threshold strong dimension of Cartesian products of graphs. This is a great opportunity for a senior undergraduate student to experience mathematical research and problem solving first-hand.
Graphs are used to model many types of networks, such as transportation systems, the link structure of websites, and data organization networks. The metric dimension of a graph can be equated to the minimum number of distance-detecting devices required to locate an intruder in the network. It is natural to ask if the number of distance-detecting devices that are required could be reduced if additional links are added to the network. This prompts the question by how much the metric dimension of a graph can be reduced by adding edges to the graph, which is the motivation for studying the threshold dimension of a graph. The problems of determining the metric dimension and the threshold dimension of a graph are NP hard, and so graph theorists focus on determining these parameters for different classes of graphs, or on finding sharp bounds. We focus on determining the threshold dimension of Cartesian products of graphs in terms of the threshold dimension of the component graphs in the product. We will also study the threshold strong dimension, which is a stronger version of the threshold dimension.
The research team will consist of myself and one undergraduate student researcher. I supervise and direct the project, but the objective is for the student to achieve some new results independently and coauthor a research paper. We begin by reviewing the relevant literature, and we draw from the techniques used by other researchers to get a good handle on what has been done and what questions remain and are of interest to researchers. The student will read about the recent work on the threshold dimension and the threshold strong dimension of a graph, and the techniques that have been used for bounding or calculating these parameters. Ideally, the student will discover some new results on these graph parameters independently. If the student has strong coding skills, they will write programs to determine the threshold and threshold strong dimension of some smaller graphs, to obtain some empirical evidence about these parameters. They will also have opportunities for professional development, helping to write up the new results we achieve for publication in a mathematical journal using LaTeX or another math text editor. They may have the opportunity to present the results of our research at an undergraduate mathematics conference or poster competition. The student will benefit academically through learning about the important interplay between groups and discrete structures such as graphs, which will reinforce and strengthen the skills needed to pursue graduate level research.
The project runs from May 19-August 21 (approximately). The successful applicant will work around 24.5 hours per week for 14 weeks, for a total of 345 hours. Some of the work will be carried out by the student independently, such as the directed readings and programming assignments, so the schedule of work is fairly flexible. We will meet regularly throughout the project to discuss the readings and our progress on the problem.
Duties:
- Perform academic research, standard tests and experiments: the student will carry out a literature review of the problem of finding the threshold dimension and threshold strong dimension of a graph.
- Monitor and document research status and findings: the student will document their progress on the problem.
- Assist with duties related to the production of articles, publications, reports and presentations: the student will write up the new results we achieve for publication in a mathematical journal using LaTeX or another math text editor. They may have the opportunity to present the results of our research at an undergraduate mathematics conference or poster competition.
- Prepare literature reviews: the student will carry out a literature search for papers relating to the problem.
- Assist in data collection, data entry, analysis and interpretation: the student may write programs to calculate the threshold dimension of smaller graphs and record the results in a data set.
Qualifications:
- Mathematics Major or Minor.
- Strong GPA in Mathematics courses.
- Completed the course Graph Theory, MATH-3401.
- Strong programming skills is an asset.
Condition(s) of Employment:
-Must be legally entitled to work in Canada.
The period of work will be fromMay 19, 2026 - August 21, 2026and the estimated total hours of work will be0-345 hours.
Note*: This position is represented by the Public Service Alliance of Canada - Research Capacity Unit.*
**Note:**The work described in this posting will be conducted in-person.
The University of Winnipeg is committed to equity, diversity and inclusion and recognizes that a diverse staff and faculty benefits and enriches the work, learning and research environments, and is essential to academic and institutional excellence. We welcome applications from all qualified individuals and encourage women, racialized persons, Indigenous persons, persons with disabilities, and 2SLGBTQ+ persons to confidentially self-identify at time of application.
The University of Winnipeg is committed to ensuring employment opportunities are accessible for all applicants. If you require accommodation supports during the recruitment process, please contact human_resources@uwinnipeg.ca.
The personal information of applicants is collected under the authority of the University of Winnipeg Act and 36(1)(b) of the Freedom of Information and Protection of Privacy Act. All personal information collected via the recruitment process is used to assess the applicant’s suitability, eligibility, and qualifications for employment, and to otherwise support recruitment activities. This information will be provided to participating members of the recruitment process. Questions regarding the collection of your personal information may be directed to the Director, HR Services, 515 Portage Avenue, Winnipeg, MB, R3B 2E9 or human_resources@uwinnipeg.ca.
About University of Winnipeg
The University of Winnipeg is noted for academic excellence, small class sizes, environmental commitment, campus diversity, Indigenous scholarship, and support of cultural arts. UWinnipeg is strongly committed to promoting access and inclusion, and is ranked 4th in Canada for reputation by Maclean's (2013). Find out more at uwinnipeg.ca. Follow us on Twitter and Facebook.
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Research Assistant, The Threshold Dimension and Threshold Strong Dimension of Cartesian Products of Graphs
About the role
Research Assistants are normally students, in a University of Winnipeg undergraduate program, employed to assist with tasks related to research. Work is performed within defined guidelines and under supervision of a more senior researcher or faculty member.
Position Overview:
The goal of this summer research project is to solve an open problem in graph theory using algebraic tools. The problem is that of determining the threshold dimension and threshold strong dimension of Cartesian products of graphs. This is a great opportunity for a senior undergraduate student to experience mathematical research and problem solving first-hand.
Graphs are used to model many types of networks, such as transportation systems, the link structure of websites, and data organization networks. The metric dimension of a graph can be equated to the minimum number of distance-detecting devices required to locate an intruder in the network. It is natural to ask if the number of distance-detecting devices that are required could be reduced if additional links are added to the network. This prompts the question by how much the metric dimension of a graph can be reduced by adding edges to the graph, which is the motivation for studying the threshold dimension of a graph. The problems of determining the metric dimension and the threshold dimension of a graph are NP hard, and so graph theorists focus on determining these parameters for different classes of graphs, or on finding sharp bounds. We focus on determining the threshold dimension of Cartesian products of graphs in terms of the threshold dimension of the component graphs in the product. We will also study the threshold strong dimension, which is a stronger version of the threshold dimension.
The research team will consist of myself and one undergraduate student researcher. I supervise and direct the project, but the objective is for the student to achieve some new results independently and coauthor a research paper. We begin by reviewing the relevant literature, and we draw from the techniques used by other researchers to get a good handle on what has been done and what questions remain and are of interest to researchers. The student will read about the recent work on the threshold dimension and the threshold strong dimension of a graph, and the techniques that have been used for bounding or calculating these parameters. Ideally, the student will discover some new results on these graph parameters independently. If the student has strong coding skills, they will write programs to determine the threshold and threshold strong dimension of some smaller graphs, to obtain some empirical evidence about these parameters. They will also have opportunities for professional development, helping to write up the new results we achieve for publication in a mathematical journal using LaTeX or another math text editor. They may have the opportunity to present the results of our research at an undergraduate mathematics conference or poster competition. The student will benefit academically through learning about the important interplay between groups and discrete structures such as graphs, which will reinforce and strengthen the skills needed to pursue graduate level research.
The project runs from May 19-August 21 (approximately). The successful applicant will work around 24.5 hours per week for 14 weeks, for a total of 345 hours. Some of the work will be carried out by the student independently, such as the directed readings and programming assignments, so the schedule of work is fairly flexible. We will meet regularly throughout the project to discuss the readings and our progress on the problem.
Duties:
- Perform academic research, standard tests and experiments: the student will carry out a literature review of the problem of finding the threshold dimension and threshold strong dimension of a graph.
- Monitor and document research status and findings: the student will document their progress on the problem.
- Assist with duties related to the production of articles, publications, reports and presentations: the student will write up the new results we achieve for publication in a mathematical journal using LaTeX or another math text editor. They may have the opportunity to present the results of our research at an undergraduate mathematics conference or poster competition.
- Prepare literature reviews: the student will carry out a literature search for papers relating to the problem.
- Assist in data collection, data entry, analysis and interpretation: the student may write programs to calculate the threshold dimension of smaller graphs and record the results in a data set.
Qualifications:
- Mathematics Major or Minor.
- Strong GPA in Mathematics courses.
- Completed the course Graph Theory, MATH-3401.
- Strong programming skills is an asset.
Condition(s) of Employment:
-Must be legally entitled to work in Canada.
The period of work will be fromMay 19, 2026 - August 21, 2026and the estimated total hours of work will be0-345 hours.
Note*: This position is represented by the Public Service Alliance of Canada - Research Capacity Unit.*
**Note:**The work described in this posting will be conducted in-person.
The University of Winnipeg is committed to equity, diversity and inclusion and recognizes that a diverse staff and faculty benefits and enriches the work, learning and research environments, and is essential to academic and institutional excellence. We welcome applications from all qualified individuals and encourage women, racialized persons, Indigenous persons, persons with disabilities, and 2SLGBTQ+ persons to confidentially self-identify at time of application.
The University of Winnipeg is committed to ensuring employment opportunities are accessible for all applicants. If you require accommodation supports during the recruitment process, please contact human_resources@uwinnipeg.ca.
The personal information of applicants is collected under the authority of the University of Winnipeg Act and 36(1)(b) of the Freedom of Information and Protection of Privacy Act. All personal information collected via the recruitment process is used to assess the applicant’s suitability, eligibility, and qualifications for employment, and to otherwise support recruitment activities. This information will be provided to participating members of the recruitment process. Questions regarding the collection of your personal information may be directed to the Director, HR Services, 515 Portage Avenue, Winnipeg, MB, R3B 2E9 or human_resources@uwinnipeg.ca.
About University of Winnipeg
The University of Winnipeg is noted for academic excellence, small class sizes, environmental commitment, campus diversity, Indigenous scholarship, and support of cultural arts. UWinnipeg is strongly committed to promoting access and inclusion, and is ranked 4th in Canada for reputation by Maclean's (2013). Find out more at uwinnipeg.ca. Follow us on Twitter and Facebook.