Abstract
A graph \(G=(V,E)\) is called supermagic if there exists a bijection \(f: E\rightarrow \{1,2,\dots ,|E|\}\) such that the weight of every vertex \(x\in V\) defined as the sum of labels f(xy) of all edges xy incident with x is equal to the same number m, called the supermagic constant.
Recently, Kovář et al. affirmatively answered a question by Madaras about existence of supermagic graphs with arbitrarily many different degrees. Their construction provided graphs with all degrees even. Therefore, they asked if there exists a supermagic graph with d different odd degrees for any positive integer d.
We answer this question in the affirmative by providing a construction based on the use of 3-dimensional magic rectangles.
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References
Hagedorn, T.R.: On the existence of magic \(n\)-dimensional rectangles. Discrete Math. 207(1–3), 53–63 (1999)
Kovář, P., Kravčenko, M., Krbeček, M., Silber, A.: Supermagic graphs with many different degrees. Discuss. Math. Graph Theory (accepted)
Zhou, C., Li, W., Zhang, Y., Su, R.: Existence of magic \(3\)-dimensional rectangles. J. Combin. Des. 26(6), 280–309 (2018)
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Froncek, D., Qiu, J. (2019). Supermagic Graphs with Many Odd Degrees. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_19
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DOI: https://doi.org/10.1007/978-3-030-25005-8_19
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