Every planar graph without 3-cycles adjacent to 4-cycles and without 6-cycles is (1, 1, 0)-colorable

Ying Bai, Xiangwen Li () and Gexin Yu
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Ying Bai: Huazhong Normal University
Xiangwen Li: Huazhong Normal University
Gexin Yu: Huazhong Normal University

Journal of Combinatorial Optimization, 2017, vol. 33, issue 4, No 11, 1354-1364

Abstract: Abstract Let $$c_{1},c_{2},\ldots ,c_{k}$$ c 1 , c 2 , … , c k be k non-negative integers. A graph G is $$(c_{1},c_{2},\ldots ,c_{k})$$ ( c 1 , c 2 , … , c k ) -colorable if the vertex set can be partitioned into k sets $$V_{1},V_{2},\ldots ,V_{k}$$ V 1 , V 2 , … , V k such that for every $$i,1\le i\le k$$ i , 1 ≤ i ≤ k , the subgraph $$G[V_{i}]$$ G [ V i ] has maximum degree at most $$c_{i}$$ c i . Steinberg (Ann Discret Math 55:211–248, 1993) conjectured that every planar graph without 4- and 5-cycles is 3-colorable. Xu and Wang (Sci Math 43:15–24, 2013) conjectured that every planar graph without 4- and 6-cycles is 3-colorable. In this paper, we prove that every planar graph without 3-cycles adjacent to 4-cycles and without 6-cycles is (1, 1, 0)-colorable, which improves the result of Xu and Wang (Sci Math 43:15–24, 2013), who proved that every planar graph without 4- and 6-cycles is (1, 1, 0)-colorable.

Keywords: Planar graphs; Improper coloring; Cycle (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-016-0039-3

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