Edge-colourings of cubic graphs and partial Steiner triple

Martin Škoviera

Comenius University, Bratislava, Slovakia

Abstract

Vizing's edge-colouring theorem divides cubic graphs into the class of 3-edge-colourable graphs (which comprises almost all of them) and a ``small" but annoying family of graphs that cannot be 3-edge-coloured. One possible approach to studying uncolourable cubic graphs consists in extending the definition of a 3-edge-colouring to include a wider class of cubic graphs. In this talk we propose a natural generalisation of the classical 3-edge-colouring based on the concept of a partial Steiner triple system. The colourings use points of the system as colours subject to the condition that any three colours meeting at a vertex form a triple. Many interesting systems occur as geometric configurations of points and lines, and the corresponding colourings seem to have a special importance.

In the talk we show that all bridgeless cubic graphs admit colourings by