List total colorings of series-parallel graphs

Xiao Zhou, Yuki Matsuo, Takao Nishizeki

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L(v)|≥min{5,Δ+1} for each vertex v and |L(e)|≥max{5,d(v)+1,d(w)+1} for each edge e=vw, where Δ is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with Δ+1 colors if Δ≥4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition.

Original languageEnglish
Pages (from-to)47-60
Number of pages14
JournalJournal of Discrete Algorithms
Issue number1
Publication statusPublished - 2005 Mar


  • Linear algorithm
  • List edge-coloring
  • List total coloring
  • NP-complete
  • Series-parallel graph


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