
TL;DR
This paper proves that any graph embeddable on a torus can be properly coloured with at most 9 colours such that each non-isolated vertex has a neighbourhood with some colour appearing an odd number of times.
Contribution
It establishes an upper bound of 9 colours for odd colourings of torus-embeddable graphs, extending understanding of graph colourings on surfaces.
Findings
Graphs on the torus admit a proper odd colouring with at most 9 colours.
The result provides a new bound in the study of odd colourings on surface-embedded graphs.
This advances the theory of graph colourings in topologically constrained settings.
Abstract
A proper vertex-colouring of a simple graph is said to be odd if, for every non-isolated vertex of , some colour appears an odd number of times in the neighbourhood of . We show that if embeds in the torus, then it admits a proper odd vertex-colouring with at most colours.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Finite Group Theory Research
