Full color theorems for L(2, 1)-colorings

Peter C. Fishburn, Fred S. Roberts

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


The span λ(G) of a graph G is the smallest k for which G's vertices can be L(2, 1)-colored, i.e., colored with integers in {0, 1,..., k} so that adjacent vertices' colors differ by at least 2, and colors of vertices at distance two differ. G is full-colorable if some such coloring uses all colors in {0, 1,..., λ(G)} and no others. We prove that all trees except stars are full-colorable. The connected graph G with the smallest number of vertices exceeding λ(G) which is not full-colorable is C6. We describe an array of other connected graphs that are not full-colorable and go into detail on full-colorability of graphs of maximum degree four or less.

Original languageEnglish (US)
Pages (from-to)428-443
Number of pages16
JournalSIAM Journal on Discrete Mathematics
Issue number2
StatePublished - 2006

ASJC Scopus subject areas

  • Mathematics(all)


  • Channel assignment problems
  • Distance-two colorings
  • No-hole colorings


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