A decomposition method on solving the linear arboricity conjecture

Abstract

AbstractA linear forest is a disjoint union of path graphs. The linear arboricity of a graph , denoted by , is the least number of linear forests into which the graph can be partitioned. Clearly, for any graph of maximum degree . For the upper bound, the long‐standing Linear Arboricity Conjecture (LAC) due to Akiyama, Exoo, and Harary from 1981 asserts that . A graph is a pseudoforest if each of its components contains at most one cycle. In this paper, we prove that the union of any two pseudoforests of maximum degree up to 3 can be decomposed into three linear forests. Combining it with a recent result of Wdowinski on the minimum number of pseudoforests into which a graph can be decomposed, we prove that the LAC holds for the following simple graph classes: ‐degenerate graphs with maximum degree , all graphs on nonnegative Euler characteristic surfaces provided the maximum degree , and graphs on negative Euler characteristic surfaces provided the maximum degree , as well as graphs with no ‐minor satisfying some conditions on maximum degrees.