On Gap-Labellings of Some Families of Graphs

30 Pages Posted: 22 Jan 2023

See all articles by Celso A. Weffort-Santos

Celso A. Weffort-Santos

affiliation not provided to SSRN

Christiane N. Campos

affiliation not provided to SSRN

Rafael C. S. Schouery

Universidade Estadual de Campinas (UNICAMP)

Abstract

This text addresses the proper gap-labelling problem in both its edge and vertex variants. Consider an assignment of integer values $\{1, \dots, k\}$ to either the edges or vertices of a graph in such a way that we obtain a proper colouring of the graph. In the edge-labelling variant, a vertex of degree at least two has its colour defined by the largest difference, or the largest ``gap'', among the labels of its incident edges; vertices of degree 1 receive as induced colour the label assigned to its (only) incident edge. For the vertex variant, the gaps are defined by the largest difference among the labels in the vertices neighbourhood, and degree-one vertices receive as colour the label assigned to their (only) neighbour. The least number $k$ of labels for which a graph admits a proper gap-labelling of its edges (vertices) is called the edge-gap (vertex-gap) number of the graph. In this article, we study both parameters, from a structural and algorithmic point of view, for traditional families of graphs such as Cycles, Crowns and Wheels. We also consider Unicyclic graphs and several families of Snarks.

Keywords: Gap-labellings, proper labellings, graph labellings

Suggested Citation

Weffort-Santos, Celso A. and Campos, Christiane N. and Schouery, Rafael C. S., On Gap-Labellings of Some Families of Graphs. Available at SSRN: https://ssrn.com/abstract=4334385 or http://dx.doi.org/10.2139/ssrn.4334385

Celso A. Weffort-Santos

affiliation not provided to SSRN ( email )

No Address Available

Christiane N. Campos

affiliation not provided to SSRN ( email )

No Address Available

Rafael C. S. Schouery (Contact Author)

Universidade Estadual de Campinas (UNICAMP) ( email )

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