Defective 2-colorings of sparse graphs

O. V. Borodin, A. V. Kostochka

Research output: Contribution to journalArticlepeer-review


A graph G is (j, k)-colorable if its vertices can be partitioned into subsets V1 and V2 such that every vertex in G[V1] has degree at most j and every vertex in G[V2] has degree at most k. We prove that if k≥2j+2, then every graph with maximum average degree at most 2(2-k+2(j+2)(k+1)) is (j, k)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close to 2(2-k+2(j+2)(k+1)) (from above) that are not (j, k)-colorable. In fact, we prove a stronger result by establishing the best possible sufficient condition for the (j, k)-colorability of a graph G in terms of the minimum, φj,k(G), of the difference φj,k(W,G)=(2-k+2(j+2)(k+1))|W|-|E(G[W])| over all subsets W of V(G). Namely, every graph G with φj,k(G)>-1k+1 is (j, k)-colorable. On the other hand, we construct infinitely many non-(j, k)-colorable graphs G with φj,k(G)=-1k+1.

Original languageEnglish (US)
Pages (from-to)72-80
Number of pages9
JournalJournal of Combinatorial Theory. Series B
Issue number1
StatePublished - Jan 2014


  • Defective coloring
  • Improper coloring
  • Maximum average degree

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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