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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University of Isfahan
2024-11-01
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| Series: | ریاضی و جامعه |
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| Online Access: | https://math-sci.ui.ac.ir/article_28265_6528a64072a5cef6776fc5d046eb8965.pdf |
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| author | Bahman Ahmadi Seyed Alireza Talebpour Shirazi Fard |
| author_facet | Bahman Ahmadi Seyed Alireza Talebpour Shirazi Fard |
| author_sort | Bahman Ahmadi |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-205548f7aea54b6489dca553d2a9fda4 |
| institution | Kabale University |
| issn | 2345-6493 2345-6507 |
| language | fas |
| publishDate | 2024-11-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | ریاضی و جامعه |
| spelling | doaj-art-205548f7aea54b6489dca553d2a9fda42025-01-05T10:12:21ZfasUniversity of Isfahanریاضی و جامعه2345-64932345-65072024-11-01938110010.22108/msci.2024.138274.159028265Relations between the distinguishing number and some other graph parametersBahman Ahmadi0Seyed Alireza Talebpour Shirazi Fard1Department of Mathematics, Shiraz University, P.O.Box 71964-84334, Shiraz, IranDepartment of Mathematics, Shiraz University, P.O.Box 71964-84334, Shiraz, IranA 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.https://math-sci.ui.ac.ir/article_28265_6528a64072a5cef6776fc5d046eb8965.pdfgraphdistinguishing numberindependence numberline graph |
| spellingShingle | Bahman Ahmadi Seyed Alireza Talebpour Shirazi Fard Relations between the distinguishing number and some other graph parameters ریاضی و جامعه graph distinguishing number independence number line graph |
| title | Relations between the distinguishing number and some other graph parameters |
| title_full | Relations between the distinguishing number and some other graph parameters |
| title_fullStr | Relations between the distinguishing number and some other graph parameters |
| title_full_unstemmed | Relations between the distinguishing number and some other graph parameters |
| title_short | Relations between the distinguishing number and some other graph parameters |
| title_sort | relations between the distinguishing number and some other graph parameters |
| topic | graph distinguishing number independence number line graph |
| url | https://math-sci.ui.ac.ir/article_28265_6528a64072a5cef6776fc5d046eb8965.pdf |
| work_keys_str_mv | AT bahmanahmadi relationsbetweenthedistinguishingnumberandsomeothergraphparameters AT seyedalirezatalebpourshirazifard relationsbetweenthedistinguishingnumberandsomeothergraphparameters |