Path decompositions and Gallai's conjecture

Let G be a connected simple graph on n vertices. Gallai's conjecture asserts that the edges of G 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 if H 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 α-operations. As a corollary, the conjecture is true if each block of H is a triangle-free graph of maximum degree at most 3.