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On the strength and independence number of powers of paths and cycles | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 27 اسفند 1402 اصل مقاله (371.63 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29087.1839 | ||
| نویسندگان | ||
| Rikio Ichishima* 1؛ Francisco Antonio Muntaner Batle2؛ Yukio Takahashi3 | ||
| 1Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan University | ||
| 2The University of Newcastle | ||
| 3Department of Electronics and Informatics, School of Science and Engineering, Kokushikan University | ||
| چکیده | ||
| A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{1,2, \ldots, n \right\}$ to the vertices of $G$. The strength $\mathrm{str}\left(G\right) $ of $G$ is defined by $\mathrm{str}\left( G\right) =\min \left\{ \mathrm{str}_{f}\left( G\right)\left\vert f\text{ is a numbering of }G\right. \right\}$, where $\mathrm{str}_{f}\left( G\right) =\max \left\{ f\left( u\right)+f\left( v\right) \left\vert uv\in E\left( G\right) \right. \right\} $. Using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. To achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles. | ||
| کلیدواژه‌ها | ||
| strength؛ independence number؛ $k$th power of a graph؛ graph labeling؛ combinatorial optimization | ||
| مراجع | ||
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[14] R. Ichishima, F.A. Muntaner-Batle, and A. Oshima, On the strength of some trees, AKCE Int. J. Graphs Comb. 17 (2020), no. 1, 486–494. https://doi.org/10.1016/j.akcej.2019.06.002 [15] R. Ichishima, F.A. Muntaner-Batle, and A. Oshima, Minimum degree conditions for the strength and bandwidth of graphs, Discrete Appl. Math. 320 (2022), 191–198. https://doi.org/10.1016/j.dam.2022.05.005 [16] R. Ichishima, F.A. Muntaner-Batle, and A. Oshima, A result on the strength of graphs by factorizations of complete graphs, Discrete Math. Lett. 8 (2022), 78–82. https://doi.org/10.47443/dml.2021.0096 [17] R. Ichishima, F.A. Muntaner-Batle, A. Oshima, and Y. Takahashi, The strength of graphs and related invariants, Memoirs Kokushikan Univ. Inf. Sci. 41 (2020), 1–8.
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