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arXiv · 2007.01431

Graphs without gap-vertex-labellings: families and bounds

C. A. Weffort-Santos·R. C. S. Schouery
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Abstract

A proper labelling of a graph $G$ is a pair $(π,c_π)$ in which $π$ is an assignment of numeric labels to some elements of $G$, and $c_π$ is a colouring induced by $π$ through some mathematical function over the set of labelled elements. In this work, we consider gap-vertex-labellings, in which the colour of a vertex is determined by a function considering the largest difference between the labels assigned to its neighbours. We present the first upper-bound for the vertex-gap number of arbitrary graphs, which is the least number of labels required to properly label a graph. We investigate families of graphs which do not admit any gap-vertex-labelling, regardless of the number of labels. Furthermore, we introduce a novel parameter associated with this labelling and provide bounds for it for complete graphs ${K_n}$.

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C. A. Weffort-Santos, R. C. S. Schouery. 2020-07-02. Graphs without gap-vertex-labellings: families and bounds. https://arxiv.org/abs/2007.01431

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