Degree-Magic Labelings on the Join and Composition of Complete Tripartite Graphs
A graph is called supermagic if there is a labeling of edges, where all edges are differently labeled with consecutive positive integers such that the sum of the labels of all edges, which are incident to each vertex of this graph, is a constant.
A graph \(G\) is called degree-magic if all edges can be labeled by integers \(1,2,\ldots ,|E(G)|\) so that the sum of the labels of the edges which are incident to any vertex \(v\) is equal to \((1+|E(G)|)\deg(v)/2\). Degree-magic graphs extend supermagic regular graphs. In this paper, the necessary and sufficient conditions for the existence of degree-magic labelings of graphs obtained by taking the join and composition of complete tripartite graphs are found.
L’. Bezegová, Balanced degree-magic complements of bipartite graphs, Discrete Math. 313 (2013), 1918 – 1923, DOI: 10.1016/j.disc.2013.05.002.
L’. Bezegová and J. Ivančo, An extension of regular supermagic graphs, Discrete Math. 310 (2010), 3571 – 3578, DOI: 10.1016/j.disc.2010.09.005.
L’. Bezegová and J. Ivančo, On conservative and supermagic graphs, Discrete Math. 311 (2011), 2428 – 2436, DOI: 10.1016/j.disc.2011.07.014.
L’. Bezegová and J. Ivančo, A characterization of complete tripartite degree-magic graphs, Discuss. Math. Graph Theory 32 (2012), 243 – 253, DOI: 10.7151/dmgt.1608.
L’. Bezegová and J. Ivančo, Number of edges in degree-magic graphs, Discrete Math. 313 (2013), 1349 – 1357, DOI: 10.1016/j.disc.2013.02.018.
J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 16 (2009), #DS6, https://www.combinatorics.org/files/Surveys/ds6/ds6v12-2009.pdf.
E. Salehi, Integer-magic spectra of cycle related graphs, Iran. J. Math. Sci. Inform. 1 (2006), 53 – 63, DOI: 10.7508/ijmsi.2006.02.004.
J. Sedláček, Theory of graphs and its applications, in: Problem 27: Proc. Symp. Smolenice Praha, (1963), 163 – 164, https://searchworks.stanford.edu/view/1365109.
B. M. Stewart, Magic graphs, Canad. J. Math. 18 (1966), 1031 – 1059, DOI: 10.4153/CJM-1966-104-7.
M. T. Varela, On barycentric-magic graphs, Iran. J. Math. Sci. Inform. 10 (2015), 121 – 129, DOI: 10.7508/ijmsi.2015.01.009.
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