$(k,1)$-coloring of sparse graphs

O.V. Borodin (1, 2) , A.O. Ivanova (3) , Mickael Montassier (4) , André Raspaud (4)

1 NSU - Novosibirsk State University
2 Sobolev Institute of Mathematics
3 Research Institute of Mathematics of Yakut - Russia
4 LaBRI - Laboratoire Bordelais de Recherche en Informatique

A graph $G$ is called {\em $(k,1)$-colorable}, if the vertex set of $G$ can be partitioned into subsets $V_1$ and $V_2$ such that the graph $G[V_1]$ induced by the vertices of $V_1$ has maximum degree at most $k$ and the graph $G[V_2]$ induced by the vertices of $V_2$ has maximum degree at most $1$. We prove that every graph with a maximum average degree less than $\frac{10k+22}{3k+9}$ admits a $(k,1)$-coloring, where $k\ge2$. In particular, every planar graph with girth at least 7 is $(2,1)$-colorable, while every planar graph with girth at least 6 is $(5,1)$-colorable. On the other hand, for each $k\ge2$ we construct non-$(k,1)$-colorable graphs whose maximum average degree is arbitrarily close to $\frac {14k}{4k+1}$.

Domaines: Mathématique discrète [cs.DM]

O.V. Borodin, A.O. Ivanova, Mickael Montassier, André Raspaud. $(k,1)$-coloring of sparse graphs. 2009. ⟨hal-00425992⟩