THE ZERO-FREE CHROMATIC POLYNOMIAL

Master of Science (Mathematics & Statistics) candidate: Katrina de Vera

26 August 2026

11:00 AM Atlantic

HSH 147

Thesis Committee:

Drs. Nancy Clarke & Iain Beaton, Supervisors
Dr. Trent Marbach, Internal Examiner
Dr. Danielle Cox, MSVU, External Examiner
Dr. Richard Karsten, Head of Math & Stats
Dr. Sazia Mahfuz, Chair of the defence

Abstract

A signed graph is a graph along with an edge signature, where the edge signature maps each edge to either +1 or −1. Signed graph colouring generalizes concepts from classical graph colouring, such as the chromatic polynomial, according to a modified definition of a proper colourings for signed graphs. In this thesis we present results on the chromatic polynomial and establish an analogous framework for the zerofree chromatic polynomial. Results include establishing closed-form expressions and bounds for the coefficients, computing the zero-free chromatic polynomial of a vertex and balanced signed K⁺_r -gluing, determining zero-free chromatically equivalent signed graphs along with their properties, ending off with directions for future research.

About Katrina…

Katrina de Vera is an M.Sc. Mathematics student supervised by Dr. Iain Beaton and Dr. Nancy Clarke. She completed her undergraduate degree at Simon Fraser University, where she conducted research under Dr. Matt deVos on the existence of Hamiltonian cycles in vertex-transitive graphs. Her current research focuses on the zero-free chromatic polynomial, a fundamental concept in signed graph colouring, which can be applied in various settings, including the Four Colour Theorem, scheduling, and resource allocation.