Graphs without gap-vertex-labellings: families and bounds
C. A. Weffort-Santos, R. C. S. Schouery

TL;DR
This paper introduces new bounds for gap-vertex-labellings in graphs, identifies graph families without such labelings, and explores a novel parameter related to this labelling method.
Contribution
It provides the first upper-bound for the vertex-gap number of arbitrary graphs and investigates graphs that do not admit gap-vertex-labellings.
Findings
First upper-bound for vertex-gap number of any graph.
Identification of graph families without gap-vertex-labellings.
Bounds established for a new parameter in complete graphs.
Abstract
A proper labelling of a graph is a pair in which is an assignment of numeric labels to some elements of , and is a colouring induced by 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 .
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