Sparse Critical Graphs for Defective (1, 3)-Coloring

Abstract A graph G is (1, 3)-colorable if its vertices can be partitioned into subsets $$V_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and $$V_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> so that every vertex in $$G[V_1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> has degree at most 1 and every vertex in $$G[V_2]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> has degree at most 3. We prove that every graph with maximum average degree at most 28/9 is (1, 3)-colorable.