First-Fit colorings of graphs with no cycles of a prescribed even length

Manouchehr Zaker () and Hossein Soltani
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Manouchehr Zaker: Institute for Advanced Studies in Basic Sciences
Hossein Soltani: Institute for Advanced Studies in Basic Sciences

Journal of Combinatorial Optimization, 2016, vol. 32, issue 3, No 8, 775-783

Abstract: Abstract The First-Fit (or Grundy) chromatic number of a graph G denoted by $$\chi _{{_\mathsf{FF}}}(G)$$ χ FF ( G ) , is the maximum number of colors used by the First-Fit (greedy) coloring algorithm when applied to G. In this paper we first show that any graph G contains a bipartite subgraph of Grundy number $$\lfloor \chi _{{_\mathsf{FF}}}(G) /2 \rfloor +1$$ ⌊ χ FF ( G ) / 2 ⌋ + 1 . Using this result we prove that for every $$t\ge 2$$ t ≥ 2 there exists a real number $$c>0$$ c > 0 such that in every graph G on n vertices and without cycles of length 2t, any First-Fit coloring of G uses at most $$cn^{1/t}$$ c n 1 / t colors. It is noted that for $$t=2$$ t = 2 this bound is the best possible. A compactness conjecture is also proposed concerning the First-Fit chromatic number involving the even girth of graphs.

Keywords: Graph coloring; First-Fit coloring algorithm; Grundy number; 05C15; 05C38; 05C35; 05C85 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-015-9900-z

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