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List strong linear 2-arboricity of sparse graphs. (English) Zbl 1218.05038
Summary: A graph $$G$$ is called $$(k, j)$$-colorable if the vertex set of $$G$$ can be partitioned into subsets $$V_{1}$$ and $$V_{2}$$ such that the graph $$G[V_{i}]$$ induced by the vertices of $$V_{i}$$ has maximum degree at most $$k$$ for $$i = 1$$ and at most $$j$$ for $$i = 2$$. In particular, F. Havet and J.-S. Sereni [J. Graph Theory 52, No. 3, 181–199 (2006; Zbl 1104.05026)] proved that each planar graph $$G$$ is list $$(1, 1)$$-colorable if its girth, $$g(G)$$, is at least 8 and list $$(2, 2)$$-colorable if $$g(G) \geq 6$$. O. V. Borodin and A. O. Ivanova [Diskretn. Anal. Issled. Oper., Ser. 1 16, No 2, 16–20(2009)] proved that every planar graph is $$(2, 1)$$-colorable if $$g(G) \geq 7$$ and $$(5, 1)$$-colorable if $$g(G) \geq 6$$. In fact, all these results were proved for each not necessarily planar sparse graph $$G$$, i.e., having a low maximum average degree, $$\text{mad}(G)$$, over all subgraphs.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C15 Coloring of graphs and hypergraphs 05C07 Vertex degrees
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