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Distinguishing chromatic number of middle and subdivision graphs | ||
| Transactions on Combinatorics | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 13 خرداد 1404 | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2025.143380.2224 | ||
| نویسندگان | ||
| Banerjee Amitayu* 1؛ Alexa Gopaulsingh2؛ Zalan Molnar2 | ||
| 1Alfred Renyi Institute of Mathematics, Realtanoda utca 13-15, 1053, Budapest, Hungary. | ||
| 2Eotvos Lorand University, Department of Logic, Muzeum krt. 4, 1088, Budapest, Hungary | ||
| چکیده | ||
| Albertson and Collins (1996) investigated the distinguishing number of a graph and the distinguishing chromatic number of a graph was introduced by Collins and Trenk (2006). In this paper, we study the distinguishing chromatic number of the middle graph and the subdivision graph of a connected graph. In particular, let G be a simple finite connected graph of order n greater than or equal to 3. We obtain the following results: (1). We apply a result of Hamada and Yoshimura from 1976 and some recent results of Alikhani and Soltani (2020) and Kalinowski and Pilsniak (2015) to determine the distinguishing chromatic number of the middle graph M(G) of the graph G. (2). In 2016, Kalinowski, Pilsniak, and Wozniak introduced and investigated the total distinguishing number D''(G) of G. Inspired by a recent result of Mirafzal (2024), we show that the distinguishing number D(S(G)) of the subdivision graph S(G) of G is D''(G). Consequently, D(S(G)) is at most $\lceil \sqrt{\Delta(G)}\rceil$. (3). We obtain a sharp upper bound for the distinguishing chromatic number of the subdivision graph S(G) of G in terms of the distinguishing number of G. | ||
| کلیدواژه‌ها | ||
| automorphism group؛ distinguishing number؛ distinguishing chromatic number؛ middle graph؛ subdivision graph | ||
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