Abstract: Gallai's conjecture asserts that every connected graph on n vertices can be decomposed into at most $\frac{{n + 1}}{2}$ paths. The E-subgraph of a graph G, denoted by $ G_e $, is the subgraph induced by the vertices of even degree in G. A triangle pair is a graph consisting of two triangles with exactly one vertex in common. In this paper, it is proved that Gallai's conjecture is true for graphs G in which $ G_e $ contains no triangle pair and each block of $ G_e $ has maximum degree at most 3.

Key words: Gallai's conjecture, Path, Triangle, Decomposition