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On twin edge colorings of graphs. (English) Zbl 1305.05068
Summary: A twin edge $$k$$-coloring of a graph $$G$$ is a proper edge coloring of $$G$$ with the elements of $$Z_{k}$$ so that the induced vertex coloring in which the color of a vertex $$v$$ in $$G$$ is the sum (in $$Z_{k}$$) of the colors of the edges incident with $$v$$ is a proper vertex coloring. The minimum $$k$$ for which $$G$$ has a twin edge $$k$$-coloring is called the twin chromatic index of $$G$$. Among the results presented are formulas for the twin chromatic index of each complete graph and each complete bipartite graph.
Reviewer: Reviewer (Berlin)

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
edge coloring; vertex coloring; factorization
Full Text:
##### References:
 [1] L. Addario-Berry, R.E.L. Aldred, K. Dalal and B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory (B) 94 (2005) 237-244. doi:10.1016/j.jctb.2005.01.001 · Zbl 1074.05031 [2] M. Anholcer, S. Cichacz and M. Milaniˇc, Group irregularity strength of connected graphs, J. Comb. Optim., to appear. doi:10.1007/s10878-013-9628-6 · Zbl 1316.05078 [3] G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer. 64 (1988) 187-192. · Zbl 0671.05060 [4] G. Chartrand, L. Lesniak and P. Zhang, Graphs & Digraphs: 5th Edition (Chapman & Hall/CRC, Boca Raton, FL, 2010). [5] G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC, Boca Raton, FL, 2008). doi:10.1201/9781584888017 [6] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 16 (2013) #DS6. · Zbl 0953.05067 [7] R.B. Gnana Jothi, Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University (1991). [8] S.W. Golomb, How to number a graph, in: Graph Theory and Computing R.C. Read (Ed.), (Academic Press, New York, 1972) 23-37. · Zbl 0293.05150 [9] R. Jones, Modular and Graceful Edge Colorings of Graphs, Ph.D. Thesis, Western Michigan University (2011). [10] R. Jones, K. Kolasinski, F. Fujie-Okamoto and P. Zhang, On modular edge-graceful graphs, Graphs Combin. 29 (2013) 901-912. doi:10.1007/s00373-012-1147-1 · Zbl 1268.05174 [11] R. Jones, K. Kolasinski and P. Zhang, A proof of the modular edge-graceful trees conjecture, J. Combin. Math. Combin. Comput. 80 (2012) 445-455. · Zbl 1247.05207 [12] M. Karoński, T. Luczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157. doi:10.1016/j.jctb.2003.12.001 · Zbl 1042.05045 [13] A. Rosa, On certain valuations of the vertices of a graph, in: Theory of Graphs, Proc. Internat. Symposium Rome 1966 (Gordon and Breach, New York 1967) 349-355. [14] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25-30 (in Russian).
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