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Every planar graph with cycles of length neither 4 nor 5 is $$(1,1,0)$$-colorable. (English) Zbl 1309.05058
Summary: Let $$d_1, d_2,\dots ,d_k$$ be $$k$$ non-negative integers. A graph $$G$$ is $$(d_1,d_2,\dots,d_k)$$-colorable, if the vertex set of $$G$$ can be partitioned into subsets $$V_1,V_2,\dots,V_k$$ such that the subgraph $$G[V_i]$$ induced by $$V_i$$ has maximum degree at most $$d_i$$ for $$i=1,2,\dots,k$$. Let $$\digamma$$ be the family of planar graphs with cycles of length neither 4 nor 5. Steinberg conjectured that every graph of $$\digamma$$ is $$(0,0,0)$$-colorable. In this paper, we prove that every graph of $$\digamma$$ is $$(1,1,0)$$-colorable.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles 05C15 Coloring of graphs and hypergraphs
##### Keywords:
planar graph; Steinberg’s conjecture; improper coloring
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##### References:
 [1] Cowen, LJ; Cowen, RH; Woodall, DR, Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J Graph Theory, 10, 187-195, (1986) · Zbl 0596.05024 [2] Eaton, N; Hull, T, Defective List colorings of planar graphs, Bull Inst Combinatorics Appl, 25, 79-87, (1999) · Zbl 0916.05026 [3] S̆krekovski R, (1999) List improper coloring of planar graphs. Combinatorics Probab Comput 8:293-299 · Zbl 0596.05024 [4] Steinberg, R, The state of the three color problem. quo vadis, graph theory?, Ann Discret Math, 55, 211-248, (1993) [5] Xu, B, On (3; 1)-coloring of plane graphs, SIAM J Discret Math, 23, 205-220, (2009) · Zbl 1221.05167
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