Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

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

1 NSU - Institute of Mathematics, NSU - Russia
2 Research Institute of Mathematics of Yakut - Russia
3 LaBRI - Laboratoire Bordelais de Recherche en Informatique
4 ALGCO - Algorithmes, Graphes et Combinatoire

A graph $G$ is $(k,0)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ every vertex has degree at most $k$, while $G[V_2]$ is edgeless. For every integer $k\ge 1$, we prove that every graph with the maximum average degree smaller than $\frac {3k+4}{k+2}$ is $(k,0)$-colorable. In particular, it follows that every planar graph with girth at least $7$ is $(8,0)$-colorable. On the other hand, we construct planar graphs with girth $6$ that are not $(k,0)$-colorable for arbitrarily large $k$.

Domaines: Mathématique discrète [cs.DM]

O.V. Borodin, A.O. Ivanova, Mickael Montassier, Pascal Ochem, André Raspaud. Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k. Journal of Graph Theory, 2010, 65 (2), pp.83-93. ⟨10.1002/jgt.20467⟩. ⟨hal-00377372⟩