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The upper domatic number of powers of graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 5، دوره 6، شماره 1، شهریور 2021، صفحه 53-65 اصل مقاله (405.37 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2020.26913.1163 | ||
| نویسندگان | ||
| Libin Chacko Samuel* 1؛ Mayamma Joseph2 | ||
| 1CHRIST (Deemed to be University), Bangalore | ||
| 2CHRIST(Deemed to be University), Bangalore | ||
| چکیده | ||
| Let $A$ and $B$ be two disjoint subsets of the vertex set $V$ of a graph $G$. The set $A$ is said to dominate $B$, denoted by $A \rightarrow B$, if for every vertex $u \in B$ there exists a vertex $v \in A$ such that $uv \in E(G)$. For any graph $G$, a partition $\pi = \{V_1,$ $V_2,$ $\ldots,$ $V_p\}$ of the vertex set $V$ is an \textit{upper domatic partition} if $V_i \rightarrow V_j$ or $V_j \rightarrow V_i$ or both for every $V_i, V_j \in \pi$, whenever $i \neq j$. The \textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. In this paper, we study the upper domatic number of powers of graphs and examine the special case when power is $2$. We also show that the upper domatic number of $k^{\mathrm{th}}$ power of a graph can be viewed as its $ k$-upper domatic number. | ||
| کلیدواژهها | ||
| Domatic number؛ $k$-domatic number؛ upper domatic partition؛ upper domatic number؛ $k$-upper domatic number | ||
| مراجع | ||
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[1] E. J. Cockayne and S. T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977), no. 3, 247–261.
[2] P. Francis and S. F. Raj, On b-coloring of powers of hypercubes, Discrete Appl. Math. 225 (2017), no. 10, 74–86.
[3] F. Harary, Graph Theory, Addison- Wesley Publishing Company, Inc, 1969.
[4] T. W. Haynes, J. T. Hedetniemi, S. T. Hedetniemi, A. McRae, and N. Phillips, The upper domatic number of a graph, AKCE Int. J. Graphs Comb. 17 (2020), no. 1, 139–148.
[5] S. T. Hedetniemi, P. Slater, and T. W. Haynes, Fundamentals of Domination in Graphs, CRC Press, 1998.
[6] L. C. Samuel and M. Joseph, New results on upper domatic number of graphs, Commun. Comb. Optim. 5 (2020), no. 2, 125–137.
[7] D. B. West, Introduction to Graph Theory, Prentice Hall, 1996.
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