Open problems of the 33rd Workshop on Cycles and Colourings
János Barát, Zdeněk Dvořák, Penny Haxell, František Kardoš, Borut Lužar, Alfréd Onderko, Jozef Rajník, Roman Soták, Nikolay Ulyanov
TL;DR
This open problems collection surveys key questions in graph coloring, flows, and connectivities arising from the 33rd Cycles and Colourings workshop. It spans the Alon–Tarsi framework for list colorability of random $4$-regular graphs, strong colouring and edge-colouring questions for degree-$2$ and claw-free graphs, and structural properties of cubic graphs and snarks, including homogeneous colorings and flow-pair conjectures. The proposals mix conjectures, partial results, and computational verifications across Ramsey-type problems, strong chromatic numbers, and almost-connectivity notions, aiming to advance understanding of 3–5 colorability, 2–4 flow interactions, and snark obstructions. Collectively, the problems illuminate deep connections between coloring, packing, and flow theories in cubic and snark-rich graphs, with potential implications for longstanding conjectures such as the $5$-flow and list-colorability thresholds.
Abstract
Since its beginnings, every Cycles and Colourings workshop holds one or two open problem sessions; this document contains the problems (together with notes regarding the current state of the art and related bibliography) presented by participants of the 33rd edition of the workshop which took place in Nový Smokovec, Slovakia during August 31st - September 5th, 2025 (see the workshop webpage https://candc.upjs.sk).