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.

中文翻译：

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走向加莱路径分解猜想

图 G 的路径分解是覆盖 G 的边集的 G 的边不相交路径的集合。 Gallai (1968) 推测 n 个顶点上的每个连通图最多允许进行一次基数的路径分解 (n+1) /2。其验证的开创性结果考虑了从 G 通过去除奇数顶点获得的图，称为 G 的 E 子图。 Lov\'asz (1968) 验证了加莱猜想的图，其 E 子图最多由一个vertex 和 Pyber (1996) 验证了其 E 子图是森林的图。2005 年，Fan 验证了 Gallai 猜想，其中 E 子图的每个块都没有三角形且最大度数最多为 3。令 calG 为 (i) 每个块的最大度数最多为 3 的图族; (ii) 每个组件要么具有最多 3 的最大度数，要么最多具有一个包含三角形的块。在本文中，我们通过验证Gallai 猜想对E 子图是calG 中图的子图的图来概括范的结果。这允许 E 子图的组件包含任意数量的三角形块，只要它们是 calG 中的图的子图。