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Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency. (English) Zbl 0596.05024
A graph is called (m,k)-colorable if its vertices can be colored with m colors such that each vertex is adjacent to at most k vertices of the same color as itself. Also (a,b)$$\geq (c,d)$$ means $$a\geq c$$ and $$b\geq d$$. Theorem A. Every outer-planar graph is (m,k)-colorable iff (m,k)$$\geq (2,2)$$ or (3,0). Theorem B. Every planar graph is (m,k)-colorable iff (m,k)$$\geq (4,0)$$ or (3,2). Theorem C. For each compact surface s, there is an integer k such that every graph embedded in s can be (4,k)-colored. Conjecture. In Theorem C, 3 can replace 4. Conjecture. If G is planar, then V(G) can be partitioned into subsets $$V_ 1$$, $$V_ 2$$, $$V_ 3$$ such that each $$v_ i$$ induces a union of disjoint paths and each $$V_ i\cup V_ j$$ induces an outerplanar graph.
Reviewer: J.Mitchem

MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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References:
 [1] Appel, Illinois J. Math. 20 pp 218– (1976) [2] Graph Theory. Addison-Wesley, Reading, MA (1969). [3] LovĂˇsz, Studia Sci. Math. Hungar. 1 pp 237– (1966) [4] and , The Appel-Haken proof of the four-color theorem. Selected Topics in Graph Theory, and , Eds. Academic, London (1978), pp. 83–101.
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