Decompositions of graphs of nonnegative characteristic with some forbidden subgraphs

Abstract

A (d,h)-decomposition of a graph G is an order pair (D,H) such that H is a subgraph of G where H has the maximum degree at most h and D is an acyclic orientation of G−E(H) of maximum out-degree at most d. A graph G is (d,h)-decomposable if G has a (d,h)-decomposition. Let G be a graph embeddable in a surface of nonnegative characteristic. It is known that if G is (d,h)-decomposable, then G is h-defective d+1-choosable. In this paper, we investigate the (d,h)-decomposable graphs and prove the following four results. (1) If G has no chord 5-cycles or no chord 6-cycles, then G is (3,1)-decomposable, which generalizes a result of Chen, Zhu and Wang [Comput. Math. Appl. 56 (2008) 2073–2078]. (2) If G has no chord 7-cycles, then G is (4,1)-decomposable, which generalizes a result of Zhang [Comment. Math. Univ. Carolin, 54 (2013) 339–344]. (3) If G has no adjacent 3-cycles and no 6-cycles and no l-cycles for l∈{5,7}, then G is (2,1)-decomposable, which improves a result of Xu and Yu [Util. Math., 76 (2008) 183–189]. (4) If G has no 4-cycles and no 8-cycles, then G is (2,1)-decomposable, which improves a result of Li, Lu, Wang and Zhu [Discrete Applied Math., 331 (2023) 147–158]. We also show that some of these results are sharp.