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Complexity of the improper twin edge coloring of graphs

Abedin, P ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1007/s00373-017-1782-7
  3. Abstract:
  4. Let G be a graph whose each component has order at least 3. Let s: E(G) → Zk for some integer k≥ 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) → Zk defined by c(v)=∑e∈Evs(e)inZk, (where the indicated sum is computed in Zk and Ev 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 χit′(G). It is known that χit′(G)=χ(G), unless χ(G)≡2(mod4) and in this case χit′(G)∈{χ(G),χ(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≥ 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 χit′(G)=χ(G) or χit′(G)=χ(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. © 2017, Springer Japan
  5. Keywords:
  6. Modular chromatic index ; NP-hardness ; Odd/even color classes ; Twin edge/vertex coloring
  7. Source: Graphs and Combinatorics ; Volume 33, Issue 4 , 2017 , Pages 595-615 ; 09110119 (ISSN)
  8. URL: https://link.springer.com/article/10.1007/s00373-017-1782-7