Relations between the distinguishing number and some other graph parameters
A distinguishing coloring of a simple graph $G$ is a vertex coloring of $G$ which is preserved only by the identity automorphism of $G$. In other words, this coloring ``breaks'' all symmetries of $G$. The distinguishing number $D(G)$ of a graph $G$ is defined to be the smallest number of c...
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| Main Authors: | , |
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| Format: | Article |
| Language: | fas |
| Published: |
University of Isfahan
2024-11-01
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| Series: | ریاضی و جامعه |
| Subjects: | |
| Online Access: | https://math-sci.ui.ac.ir/article_28265_6528a64072a5cef6776fc5d046eb8965.pdf |
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| Summary: | A distinguishing coloring of a simple graph $G$ is a vertex coloring of $G$ which is preserved only by the identity automorphism of $G$. In other words, this coloring ``breaks'' all symmetries of $G$. The distinguishing number $D(G)$ of a graph $G$ is defined to be the smallest number of colors in a distinguishing coloring of $G$. This concept of “symmetry breaking” was first proposed by Babai in 1977 and after the publication of a seminal paper by Albertson in 1996, it attracted the attention of many mathematicians. In this paper, along with studying some relations between $D(G)$ and some other important graph parameters, we introduce the concept of a $(D,\alpha)$-ordinary graph which displays the comparison between $D(G)$ and the independence number $\alpha(G)$. Then we consider the classification of $(D,\alpha)$-ordinary graphs in various families of graphs such as forests, cycles, generalized Johnson graphs, Cartesian products of graphs and line graphs of connected claw-free graphs. We also propose some conjectures and discuss about some future research directions in this topic. |
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| ISSN: | 2345-6493 2345-6507 |