The summer of 2014 was a defining period for me. Thanks to funding from the University of Warwick and the Czech Science Foundation, I had the unique opportunity to study under Professor Daniel Kráľ, delving into the intricacies of plane graph colourings. My academic journey led me to both Charles University in Prague and the University of West Bohemia in Pilsen. Daniel Kráľ was instrumental in orchestrating this adventure, turning the summer into a cherished memory. Marking my maiden voyage traveling solo, this summer was nothing short of transformative!

The quintessential plane graph colouring challenge is encapsulated by the renowned 4-colour theorem. Simply put, this theorem posits that any flat map divided into regions can be coloured using just four distinct colours, ensuring that no adjacent regions share the same colour. Interestingly, four is the bare minimum to achieve this feat across all maps. While this theorem may sound straightforward, its proof is anything but. It was eventually unravelled with the aid of computer-assisted proofs, marking a pioneering approach to problem-solving that ignited considerable debate. For the more Mathematically inclined, the theorem's formal statement is a plane graph — one embeddable in 2-dimensional Euclidean space — can have its vertices coloured using 4 distinct colours in such a way that adjacent vertices don't use identical colours.

My research concentrated on an intriguing variant of this problem, emphasizing facial constraints. What if, in addition to ensuring non-adjacent vertices differed in colour, we also mandated that every face of the graph should sport a unique dominant shade? Such constraints have can have applications in algorithmic studies. My efforts culminated in the discovery that five distinct colours would suffice in such scenarios. These findings formed the backbone of my paper Coloring of Plane Graphs with Unique Maximal Colors on Faces, which was published just after. Later, in 2018, Bernard Lidický, Kacy Messerschmidt, and Riste Škrekovski substantiated that the constraint of five colours was indeed the optimal solution in their paper A counterexample to a conjecture on facial unique-maximal colorings. Their proof elucidated this by designing a graph that necessitated 5 colours.

Reflecting on that summer, it was a cascade of 'firsts' for me: my inaugural solo journey, my initiation into the mathematical publishing sphere, and my debut academic sojourn to an esteemed global institution.