Towards Gallai's path decomposition conjecture

Abstract

AbstractA path decomposition of a graph is a collection of edge‐disjoint paths of that covers the edge set of . Gallai conjectured that every connected graph on vertices admits a path decomposition of cardinality at most . Seminal results toward its verification consider the graph obtained from by removing its vertices of odd degree, which is called the E‐subgraph of . Lovász verified Gallai's Conjecture for graphs whose E‐subgraphs consist of at most one vertex, and Pyber verified it for graphs whose E‐subgraphs are forests. In 2005, Fan verified Gallai's Conjecture for graphs in which each block, that is, each maximal 2‐connected subgraph, of their E‐subgraph is triangle‐free and has maximum degree at most 3. Let be the family of graphs for which (a) each block has maximum degree at most ; and (b) 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 . 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 .