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Group sum chromatic number of graphs. (English) Zbl 1333.05100
Summary: We investigate the group sum chromatic number ($$\chi_g^\varSigma(G)$$) of graphs, i.e. the smallest value $$s$$ such that taking any abelian group $$\mathcal{G}$$ of order $$s$$, there exists a function $$f : E(G) \to \mathcal{G}$$ such that the sums of edge labels properly colour the vertices. It is known that $$\chi_g^\varSigma(G) \in \{\chi(G), \chi(G) + 1 \}$$ for any graph $$G$$ with no component of order less than $$3$$ and we characterize the graphs for which $$\chi_g^\varSigma(G) = \chi(G)$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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##### References:
 [1] Addario-Berry, L.; Dalal, K.; McDiarmid, C.; Reed, B. A.; Thomason, A., Vertex-colouring edge-weightings, Combinatorica, 27, 1-12, (2007) · Zbl 1127.05034 [2] Addario-Berry, L.; Dalal, K.; Reed, B. A., Degree constrainted subgraphs, Discrete Appl. Math., 156, 1168-1174, (2008) · Zbl 1147.05055 [3] Beals, R.; Gallian, J.; Headley, P.; Jungreis, D., Harmonious groups, J. Combin. Theory Ser. A, 56, 223-238, (1991) · Zbl 0718.20013 [4] Cavenagh, N.; Combe, D.; Nelson, A. M., Edge-magic group labellings of countable graphs, Electron. J. Combin., 13, #R92, (2006), 19 pp · Zbl 1111.05085 [5] Combe, D.; Nelson, A. M.; Palmer, W. D., Magic labellings of graphs over finite abelian groups, Australas. J. Combin., 29, 259-271, (2004) · Zbl 1050.05107 [6] Froncek, D., Group distance magic labeling of Cartesian products of cycles, Australas. J. Combin., 55, 167-174, (2013) · Zbl 1278.05210 [7] Gallian, J. A., A dynamic survey of graph labeling, Electron. J. Combin., (2015), DS6,http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6 [8] Graham, R. L.; Sloane, N. J.A., On additive bases and harmonious graphs, SIAM J. Algebr. Discrete Methods, 1, 382-404, (1980) · Zbl 0499.05049 [9] Hovey, M., $$A$$-cordial graphs, Discrete Math., 93, 183-194, (1991) · Zbl 0753.05059 [10] Kalkowski, M., A note on 1,2-conjecture, (2009), (Ph.D. Thesis) [11] Kalkowski, M.; Karoński, M.; Pfender, F., Vertex-coloring edge-weightings: towards the 1-2-3-conjecture, J. Combin. Theory Ser. B, 100, 347-349, (2010) · Zbl 1209.05087 [12] Kaplan, G.; Lev, A.; Roditty, Y., On zero-sum partitions and anti-magic trees, Discrete Math., 309, 2010-2014, (2009) · Zbl 1229.05031 [13] Karoński, M.; Łuczak, T.; Thomason, A., Edge weights and vertex colours, J. Combin. Theory Ser. B, 91, 151-157, (2004) · Zbl 1042.05045 [14] Lu, H.; Yu, Q.; Zhang, C.-Q., Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32, 21-27, (2011) · Zbl 1203.05056 [15] Stanley, R. P., Linear homogeneous Diophantine equations and magic labelings of graphs, Duke Math. J., 40, 607-632, (1973) · Zbl 0269.05109 [16] Wang, T.; Yu, Q., On vertex-coloring 13-edge-weighting, Front. Math. China, 3, 581-587, (2008) · Zbl 1191.05048 [17] Żak, A., Harmonious orders of graphs, Discrete Math., 309, 6055-6064, (2009) · Zbl 1188.05138
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