Total colorings of degenerate graphs

Shuji Isobe, Xiao Zhou, Takao Nishizeki

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)167-182
Number of pages16
JournalCombinatorica
Volume27
Issue number2
DOIs
Publication statusPublished - 2007 Mar 1

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics

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