arXiv:1403.6288v1 [math.CO] 25 Mar 2014 The Complexity of the Sigma Chromatic Number of Cubic Graphs Ali Dehghan a , Mohammad-Reza Sadeghi a , Arash Ahadi b a Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran b Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran ∗ Abstract The sigma chromatic number of a graph G, denoted by σ(G), is the minimum number k that the vertices can be partitioned into k disjoint sets V 1 ,...,V k such that for every two adjacent vertices u and v there is an index i that u and v have different numbers of neighbors in V i . We show that, it is NP-complete to decide for a given 3-regular graph G, whether σ(G) = 2. Also, we prove that for every k ≥ 3, it is NP-complete to decide whether σ(G)= k for a given graph G. Furthermore, for planar 3-regular graphs with σ = 2, we show that the problem of minimizing the size of a set is NP-hard. Key words: Sigma coloring; Lucky labeling; Additive coloring; Computational Complexity; Planar Not-All-Equal 3-Sat; Planar Not-All-Equal 3-Sat Type 2. Subject classification: 05C15, 05C20, 68Q25 1 Introduction In 2004, Karo´ nski,  Luczak and Thomason introduced a new coloring of a graph which is generated via edge labeling [12]. Let f : E(G) → N be a labeling of the edges of a graph G by positive integers such that for every two adjacent vertices v and u, S (v) = S (u), where S (v) denotes the sum of labels of all edges incident with v. It is conjectured that three integer labels {1, 2, 3} are sufficient for every connected graph, except K 2 [12]. Currently the best bound is 5 [11]. It is shown that determining whether a given graph has a weighting of the edges from {1, 2} that induces a proper vertex coloring is NP-complete [8]. Recently, it was shown that for a given 3-regular graph has a weighting of the edges from {a, b},(a = b) that induces a proper vertex coloring is NP-complete [7]. ∗ E-mail addresses: ali dehghan16@aut.ac.ir, msadeghi@aut.ac.ir, arash ahadi@mehr.sharif .edu.. 1