Towards Gallai's path decomposition conjecture

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most (n+1)/2. Seminal results towards its verification consider the graph obtained from G by removing its vertices of odd degree, which is called the E-subgraph of G. Lov\'asz (1968) verified Gallai's Conjecture for graphs whose E-subgraphs consist of at most one vertex, and Pyber (1996) verified it for graphs whose E-subgraphs are forests. In 2005, Fan verified Gallai's Conjecture for graphs in which each block of their E-subgraph is triangle-free and has maximum degree at most 3. Let calG be the family of graphs for which (i) each block has maximum degree at most 3; and (ii) each component either has maximum degree at most 3 or has at most one block that contains triangles. In this paper, we generalize Fan's result by verifying Gallai's Conjecture for graphs whose E-subgraphs are subgraphs of graphs in calG. This allows the components of the E-subgraphs to contain any number of blocks with triangles as long as they are subgraphs of graphs in calG.