Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.

翻译：Gallai路径分解猜想断言：若$G$是$n$个顶点的连通图，则$G$的边可分解为至多$\lceil \frac{n}{2} \rceil$条路径。若一个图可由$2k+1$个顶点的完全图删除至多$k-1$条边得到，则称该图为奇半团。Bonamy与Perrett提出：所有$n$个顶点的连通图$G$的边是否均可分解为至多$\lfloor \frac{n}{2} \rfloor$条路径，除非$G$是奇半团。若图$G$的每个子图均存在度数不超过2的顶点，则称$G$为2-退化图。本文证明：任何$n$个顶点的连通2-退化图$G$的边均可分解为至多$\lfloor \frac{n}{2} \rfloor$条路径，除非$G$是三角形。