Total Colorings Of Degenerate Graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤s. We prove that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.

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Correspondence to Shuji Isobe.

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Isobe, S., Zhou, X. & Nishizeki, T. Total Colorings Of Degenerate Graphs. Combinatorica 27, 167–182 (2007). https://doi.org/10.1007/s00493-007-0050-5

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Mathematics Subject Classification (2000):

  • 05C15
  • 05C85
  • 90C39