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$$(1,k)$$-coloring of graphs with girth at least five on a surface. (English) Zbl 1359.05099
Summary: A graph is $$(d_1\ldots,d_r)$$-colorable if its vertex set can be partitioned into $$r$$ sets $$V_1,\ldots,V_r$$ so that the maximum degree of the graph induced by $$V_i$$ is at most $$d_i$$ for each $$i\in\{1,\ldots,r\}$$. For a given pair $$(g,d_1)$$, the question of determining the minimum $$d_2=d_2(g,d_1)$$ such that planar graphs with girth at least $$g$$ are $$(d_1,d_2)$$-colorable has attracted much interest. The finiteness of $$d_2(g,d_1)$$ was known for all cases except when $$(g,d_1)=(5,1)$$. M. Montassier and P. Ochem [Electron. J. Comb. 22, No. 1, Research Paper P1.57, 13 p. (2015; Zbl 1308.05052)] explicitly asked if $$d_2(5,1)$$ is finite. We answer this question in the affirmative with $$d_2(5,1)\leq10$$; namely, we prove that all planar graphs with girth at least five are $$(1,10)$$-colorable. Moreover, our proof extends to the statement that for any surface $$S$$ of Euler genus $$\gamma$$, there exists a $$K=K(\gamma)$$ where graphs with girth at least five that are embeddable on $$S$$ are $$(1,K)$$-colorable. On the other hand, there is no finite $$k$$ where planar graphs (and thus embeddable on any surface) with girth at least five are $$(0,k)$$-colorable.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C07 Vertex degrees 05C15 Coloring of graphs and hypergraphs
##### Keywords:
improper coloring; discharging; graphs on surfaces
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