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f-Colorings of some graphs of f-class 1

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Abstract

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v ∈ V(G) at most f(v) times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by x′ f (G). Any simple graph G has the f-chromatic index equal to Δ f (G) or Δ f (G) + 1, where Δ f (G) = max v∈V(G){⌈d(v)/f(v)⌊}. If x′ f (G) = Δ f (G), then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with Δ f (G) colors are given.

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Authors and Affiliations

  1. College of Mathematics Science, Shandong Normal University, Ji’nan, 250014, P. R. China

    Xia Zhang

  2. School of Mathematics and System Sciences, Shandong University, Ji’nan, 250100, P. R. China

    Xia Zhang & Gui Zhen Liu

Authors
  1. Xia Zhang
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  2. Gui Zhen Liu
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Corresponding author

Correspondence to Xia Zhang.

Additional information

This research is supported by NSFC(10471078, 60673047), RSDP(20040422004) and NSF of Hebei(A2007000002) of China

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Zhang, X., Liu, G.Z. f-Colorings of some graphs of f-class 1. Acta. Math. Sin.-English Ser. 24, 743–748 (2008). https://doi.org/10.1007/s10114-007-6194-9

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  • DOI: https://doi.org/10.1007/s10114-007-6194-9

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