The problem of path decomposition for graphs with treewidth at most 4

Changhong Lu, Niping Yi

Abstract: Gallai conjectured that every connected graph with n vertices admits a decomposition into at most ⌈n2⌉ paths. The conjecture has been proved for some special cases. Recently, Botler et al. (2020) proved that Gallai's conjecture holds for graphs with treewidth at most 3. In this paper, we show that if G is a graph with treewidth at most 4, then G can be decomposed into at most ⌊n2⌋ paths or G is isomorphic to K3, to K5−(K5−e), or to K5.