Abstract
In a graph \(G=(V,E)\), each vertex \(v\in V\) is assigned 0, 1 or 2 such that each vertex assigned 0 is adjacent to at least one vertex assigned 2 or two vertices assigned 1. Such an assignment is called an Italian dominating function (IDF) of G. The weight of an IDF f is \(w(f)=\sum _{v\in V}f(v)\). The Italian domination number of G is \(\gamma _{I}(G)=\min _{f} w(f)\). In this paper, we investigate the Italian domination number of the Cartesian product of paths, \(P_n\Box P_m\). We obtain the exact values of \(\gamma _{I}(P_n\Box P_2)\) and \(\gamma _{I}(P_n\Box P_3)\). Also, we present a bound of \(\gamma _{I}(P_n\Box P_m)\) for \(m\ge 4\), that is \(\frac{mn}{3}+\frac{m+n-4}{9}\le \gamma _{I}(P_{n}\Box P_{m})\le \frac{mn+2m+2n-8}{3}\) where the lower bound is improved since the general lower bound is \(\frac{mn}{3}\) presented by Chellali et al. (Discrete Appl Math 204:22–28, 2016). By the results of this paper, together with existing results, we give \(P_n\Box P_2\) and \(P_n\Box P_3\) are examples for which \(\gamma _{I}=\gamma _{r2}\) where \(\gamma _{r2}\) is the 2-rainbow domination number. This can partially solve the open problem presented by Brešar et al. (Discrete Appl Math 155:2394–2400, 2007). Finally, Vizing’s conjecture on Italian domination in \(P_n\Box P_m\) is checked.
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References
Amjadi J, Asgharsharghi L, Dehgardi N, Furuya M, Sheikholeslami SM, Volkmann L (2017) The \(k\)-rainbow reinforcement numbers in graphs. Discrete Appl Math 217:394–494
Brešar B, Šumenjak TK (2007) On the 2-rainbow domination in graphs. Discrete Appl Math 155:2394–2400
Brešar B, Henning MA, Rall DF (2008) Rainbow domination in graphs. Taiwan J Math 12(1):213–225
Chellali M, Haynes TW, Hedetniemi ST, McRae AA (2016) Roman 2-domination. Discrete Appl Math 204:22–28
Cockayne EJ, Dreyer PA, Hedetniemi SM, Hedetniemi ST (2004) Roman domination in graphs. Discrete Math 278:11–22
Fan WJ, Ye AS, Miao F, Shao ZH, Samodivkin V, Sheikholeslami SM (2019) Outer-independent Italian domination in graphs. IEEE Access 7:22756–22762
Gao H, Xu TT, Yang YS (2019) Bagging approach for Italian domination in \(C_n\Box P_m\). IEEE Access 7:105224–105234
Gao H, Wang PH, Liu EM, Yang YS (2020) More results on Italian domination in \(C_{n}\Box C_{m}\). Mathematics 8:465
Hao GL, Hu KX, Wei SL, Xu ZJ (2018) Global Italian domination in graphs. Quaest Math 41:1–15
Haynes TW, Henning MA (2019) Perfect Italian domination in trees. Discrete Appl Math 260:164–177
Henning MA, Klostermeyer WF (2017) Italian domination in trees. Discrete Appl Math 217:557–564
Li ZP, Shao ZH, Xu J (2018) Weak 2-domination number of Cartesian products of cycles. J Comb Optim 35(1):75–85
Rahmouni A, Chellali M (2018) Independent Roman 2-domination in graphs. Discrete Appl Math 236:408–414
Stȩpień Z, Szymaszkiewicz A, Szymaszkiewicz L, Zwierzchowski M (2014) 2-rainbow domination number of \(C_{n}\Box C_{5}\). Discrete Appl Math 170:113–116
Vizing VG (1968) Some unsolved problems in graph theory. Uspehi Mater Nauk 23:117–134
Wang Y, Wu XL, Dehgardi N, Amjadi J, Khoeilar R, Liu JB (2019) \(k\)-rainbow domination number of \(P_{3}\Box P_{n}\). Mathematics 7:203
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The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which will greatly improve the presentation.
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This work is supported by National Natural Science Foundation of China (NSFC), Grand No. is 61562066.
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Hong Gao contributes for methodology, project administration and the final draft. Tingting Feng contributes for resources, some computations and wrote the initial draft of the paper. Yuansheng Yang contributes for supervision and formal analysing.
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Gao, H., Feng, T. & Yang, Y. Italian domination in the Cartesian product of paths. J Comb Optim 41, 526–543 (2021). https://doi.org/10.1007/s10878-020-00694-x
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DOI: https://doi.org/10.1007/s10878-020-00694-x