Path decompositions and Gallai's conjecture

LetG be a connected simple graph on n vertices. Gallai’s conjecture asserts that the edges ofG can be decomposed into n2 paths. Let H be the subgraph induced by the vertices of even degree in G. Lovász showed that the conjecture is true ifH contains at most one vertex. Extending Lovász’s result, Pyber proved that the conjecture is true if H is a forest. A forest can be regarded as a graph in which each block is an isolated vertex or a single edge (and so each block has maximum degree at most 1). In this paper, we show that the conjecture is true if H can be obtained from the emptyset by a series of so-defined -operations. As a corollary, the conjecture is true if each block of H is a triangle-free graph of maximum degree at most 3. © 2004 Elsevier Inc. All rights reserved.