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computational complexity

, Volume 14, Issue 4, pp 308–340 | Cite as

The complexity of chromatic strength and chromatic edge strength

  • Dániel Marx
Original Paper

Abstract.

The sum of a coloring is the sum of the colors assigned to the vertices (assuming that the colors are positive integers). The sum ∑ (G) of graph G is the smallest sum that can be achieved by a proper vertex coloring of G. The chromatic strength s(G) of G is the minimum number of colors that is required by a coloring with sum ∑ (G). For every k, we determine the complexity of the question “Is s(G) ≤ k?”: it is coNP-complete for k = 2 and Θ 2 p -complete for every fixed k ≥ 3. We also study the complexity of the edge coloring version of the problem, with analogous definitions for the edge sum ∑′(G) and the chromatic edge strength s′(G). We show that for every k ≥ 3, it is Θ 2 p -complete to decide whether s′(G) ≤ k. As a first step of the proof, we present graphs for every r ≥ 3 with chromatic index r and edge strength r + 1. For some values of r, such graphs have not been known before.

Keywords.

Graph coloring chromatic strength chromatic number chromatic index 

Subject classification.

68Q17 

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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Department of Computer Science and Information TheoryBudapest University of Technology and EconomicsBudapestHungary

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