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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
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References:
 [1] Appel, Every planar map is four colorable. I. Discharging, Illinois J Math 21 (3) pp 429– (1977) · Zbl 0387.05009 [2] Appel, Every planar map is four colorable. II. Reducibility, Illinois J Math 21 (3) pp 491– (1977) · Zbl 0387.05010 [3] Borodin, Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k, J Graph Theory 65 (2) pp 83– (2010) · Zbl 1209.05177 · doi:10.1002/jgt.20467 [4] Borodin, (k,j)-coloring of sparse graphs, Discrete Appl Math 159 (17) pp 1947– (2011) · Zbl 1239.05059 · doi:10.1016/j.dam.2011.06.021 [5] Borodin, (k, 1)-coloring of sparse graphs, Discrete Math 312 (6) pp 1128– (2012) · Zbl 1238.05084 · doi:10.1016/j.disc.2011.11.031 [6] Borodin, On 1-improper 2-coloring of sparse graphs, Discrete Math 313 (22) pp 2638– (2013) · Zbl 1281.05060 · doi:10.1016/j.disc.2013.07.014 [7] Borodin, Vertex decompositions of sparse graphs into an independent set and a subgraph of maximum degree at most 1, Sibirsk Mat Zh 52 (5) pp 1004– (2011) · Zbl 1232.05073 [8] Borodin, Defective 2-colorings of sparse graphs, J Combin Theory Ser B 104 pp 72– (2014) · Zbl 1282.05041 · doi:10.1016/j.jctb.2013.10.002 [9] Borodin, Near proper 2-coloring the vertices of sparse graphs, Diskretn Anal Issled Oper 16 (2) pp 16– (2009) [10] I. Choi A. Raspaud Planar graphs with minimum cycle length at least 5 are (3, 5)-colorable 338 2015 [11] Cowen, Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J Graph Theory 10 (2) pp 187– (1986) · Zbl 0596.05024 · doi:10.1002/jgt.3190100207 [12] Eaton, Defective list colorings of planar graphs, Bull Inst Combin Appl 25 pp 79– (1999) · Zbl 0916.05026 [13] Esperet, A complexity dichotomy for the coloring of sparse graphs, J Graph Theory 73 (1) pp 85– (2013) · Zbl 1264.05049 · doi:10.1002/jgt.21659 [14] Glebov, Path partitions of planar graphs, Sib Èlektron Mat Izv 4 pp 450– (2007) · Zbl 1132.05315 [15] Havet, Improper choosability of graphs and maximum average degree, J Graph Theory 52 (3) pp 181– (2006) · Zbl 1104.05026 · doi:10.1002/jgt.20155 [16] J. Kim A. V. Kostochka X. Zhu Private communication 2013 [17] Montassier, Near-colorings: non-colorable graphs and NP-completeness, Electron J Combin 22 (1) (2015) · Zbl 1308.05052 [18] Škrekovski, List improper colourings of planar graphs, Combin Probab Comput 8 (3) pp 293– (1999) · Zbl 0940.05031 · doi:10.1017/S0963548399003752 [19] Škrekovski, List improper colorings of planar graphs with prescribed girth, Discrete Math 214 (1-3) pp 221– (2000) · Zbl 0940.05027 · doi:10.1016/S0012-365X(99)00145-4
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