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Complexity of the improper twin edge coloring of graphs. (English) Zbl 1371.05067
Summary: Let $$G$$ be a graph whose each component has order at least 3. Let $$s : E(G) \rightarrow \mathbb {Z}_k$$ for some integer $$k\geq 2$$ be an improper edge coloring of $$G$$ (where adjacent edges may be assigned the same color). If the induced vertex coloring $$c : V (G) \rightarrow \mathbb {Z}_k$$ defined by $$c(v) = \sum_{e\in E_v} s(e)$$ in $$\mathbb {Z}_k$$, (where the indicated sum is computed in $$\mathbb {Z}_k$$ and $$E_v$$ denotes the set of all edges incident to $$v$$) results in a proper vertex coloring of $$G$$, then we refer to such a coloring as an improper twin $$k$$-edge coloring. The minimum $$k$$ for which $$G$$ has an improper twin $$k$$-edge coloring is called the improper twin chromatic index of $$G$$ and is denoted by $$\chi^\prime_{it}(G)$$. It is known that $$\chi^\prime_{it}(G)=\chi (G)$$, unless $$\chi (G) \equiv 2 \pmod 4$$ and in this case $$\chi^\prime_{it}(G)\in \{\chi (G), \chi (G)+1\}$$. In this paper, we first give a short proof of this result and we show that if $$G$$ admits an improper twin $$k$$-edge coloring for some positive integer $$k$$, then $$G$$ admits an improper twin $$t$$-edge coloring for all $$t\geq k$$; we call this the monotonicity property. In addition, we provide a linear time algorithm to construct an improper twin edge coloring using at most $$k+1$$ colors, whenever a $$k$$-vertex coloring is given. Then we investigate, to the best of our knowledge the first time in literature, the complexity of deciding whether $$\chi^\prime_{it}(G)=\chi (G)$$ or $$\chi^\prime_{it}(G)=\chi (G)+1$$, and we show that, just like in case of the edge chromatic index, it is NP-hard even in some restricted cases. Lastly, we exhibit several classes of graphs for which the problem is polynomial.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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##### References:
 [1] Addario-Berry, L; Aldred, REL; Dalal, K; Reed, BA, Vertex colouring edge partitions, J. Comb. Theory (B), 94, 237-244, (2005) · Zbl 1074.05031 [2] Andrews, E; Helenius, L; Johnston, D; VerWys, J; Zhang, Ping, On twin edge colorings of graphs, Discuss. Math. Graph Theory, 34, 613-627, (2014) · Zbl 1305.05068 [3] Anholcer, M; Cichacz, S, Group sum chromatic number of graphs, Eur. J. Comb., 55, 73-81, (2016) · Zbl 1333.05100 [4] Anholcer, M., Cichacz, S., Milanic̆, M.: Group irregularity strength of connected graphs. J. Comb. Optim. 30(1), 1-17 (2015) · Zbl 1316.05078 [5] Appel, K; Haken, W, The solution of the four-color map problem, Sci. Amer., 237, 108-121, (1977) [6] Brooks, RL, On colouring the nodes of a network, Math. Proc. Camb. Philos. Soc., 37, 194-197, (1941) · Zbl 0027.26403 [7] Chartrand, G., Zhang, P.: Chromatic Graph Theory. CRC Press, Boca Raton (2008) · Zbl 1169.05001 [8] Clarke, G., Demange, M., Roshchina, V.: Lecture Notes-Discrete Mathematics, RMIT University [9] Edwards, K; Hornák, M; Wozniak, M, On the neighbour-distinguishing index of a graph, Graphs Comb., 22, 341-350, (2006) · Zbl 1107.05032 [10] Flandrin, E; Marczyk, A; Przybyło, J; Saclé, J-F; Woźniak, M, Neighbor sum distinguishing index, Graphs Comb., 29, 1329-1336, (2013) · Zbl 1272.05047 [11] Garey, M.R., Johnson, D.S.: Computers and Intractability, a Guide to the Theory of $$\cal{NP}$$-Completeness. Freeman, New York (1979) · Zbl 0411.68039 [12] Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, Computer Science and Applied Mathematics. Academic Press, New York (1980) · Zbl 0541.05054 [13] Jones, R; Kolasinski, K; Okamoto, F; Zhang, P, Modular neighbor-distinguishing edge colorings of graphs, J. Combin. Math. Combin. Comput., 76, 159-175, (2011) · Zbl 1233.05104 [14] Jones, R; Kolasinski, K; Okamoto, F; Zhang, P, On modular chromatic indexes of graphs, J. Combin. Math. Combin. Comput., 82, 295-306, (2012) · Zbl 1251.05057 [15] Karonski, M; Luczak, T; Thomason, A, Edge weights and vertex colours, J. Combin. Theory (B), 91, 151-157, (2004) · Zbl 1042.05045 [16] Kratsch, D; Stewart, L, Domination on cocomparability graphs, SIAM J. Discret. Math., 6, 400-417, (1993) · Zbl 0780.05032 [17] Robertson, N., Sanders, D.P., Seymour, P., Thomas, R.: Efficiently four-coloring planar graphs. In: Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pp. 571-575, ACM (1996) · Zbl 0917.05030 [18] Seamone, B.: The 1-2-3 Conjecture and related problems: a survey. arXiv:1211.5122 [math.CO] (2012) (Preprint) [19] Zhang, P.: Color-Induced Graph Colorings. Springer, Berlin (2015) · Zbl 1365.05006
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