Adjacent Vertex Distinguishing Coloring of Fuzzy Graphs

Zengtai Gong () and Chen Zhang
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Zengtai Gong: College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
Chen Zhang: College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Mathematics, 2023, vol. 11, issue 10, 1-25

Abstract: In this paper, we consider the adjacent vertex distinguishing proper edge coloring (for short, AVDPEC) and the adjacent vertex distinguishing total coloring (for short, AVDTC) of a fuzzy graph. Firstly, this paper describes the development process, the application areas, and the existing review research of fuzzy graphs and adjacent vertex distinguishing coloring of crisp graphs. Secondly, we briefly introduce the coloring theory of crisp graphs and the related theoretical basis of fuzzy graphs, and add some new classes of fuzzy graphs. Then, based on the α -cuts of fuzzy graphs and distance functions, we give two definitions of the AVDPEC of fuzzy graphs, respectively. A lower bound on the chromatic number of the AVDPEC of a fuzzy graph is obtained. With examples, we show that some results of the AVDPEC of a crisp graph do not carry over to our set up; the adjacent vertex distinguishing chromatic number of the fuzzy graph is different from the general chromatic number of a fuzzy graph. We also give a simple algorithm to construct a ( d , f ) -extended AVDPEC for fuzzy graphs. After that, in a similar way, two definitions of the AVDTC of fuzzy graphs are discussed. Finally, the future research directions of distinguishing coloring of fuzzy graphs are given.

Keywords: fuzzy set; fuzzy graph; adjacent vertex distinguishing proper edge coloring; adjacent vertex distinguishing total coloring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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