Abstract
We study a (k + 1)-coloring problem in a class of (k,s)-dart graphs, k,s ≥ 2, where each vertex of degree at least k + 2 belongs to a (k,i)-diamond, i ≤ s. We prove that dichotomy holds, that means the problem is either NP-complete (if k < s), or can be solved in linear time (if k ≥ s). In particular, in the latter case we generalize the classical Brooks Theorem, that means we prove that a (k, s)-dart graph, k ≥ max {2,s}, is (k + 1)-colorable unless it contains a component isomorphic to K k + 2.
Supported by grants VEGA 2/0118/10 and ARRS Research Program P1-0297.
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Kochol, M., Škrekovski, R. (2011). Dichotomy for Coloring of Dart Graphs. In: Iliopoulos, C.S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2010. Lecture Notes in Computer Science, vol 6460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19222-7_9
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DOI: https://doi.org/10.1007/978-3-642-19222-7_9
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