Based on data from the International Federation of Robotics (IFR) and Chinese A-Share Listed Companies 2011–2019, this paper evaluates how the penetration of industrial robots in China affects the labor income share from both theoretical and empirical perspectives. We first develop a theoretical framework considering robots in the task model, consider different tasks matching different types of labor, construct a labor income share model including robots based on the task model, and perform a theoretical analysis finding that industrial robots can improve labor income. Then, we explore the mechanism of labor price distortion as an intermediate variable that can reduce the labor income share, finding that robots are able to effectively reduce the negative impact of labor price distortions on labor income share. After correcting for potential endogeneity problems, the results confirm a positive, significant, and lasting impact of industrial robot penetration on labor income share, confirming that labor price distortion is the underlying mechanisms through which the robots can effectively reduce the negative impact of labor price distortions on labor income share. At the same time, the impact of robots on firms’ labor income shares varies significantly across different regions, external financial dependence, and skill premiums. Our findings can help the government to provide a decision-making basis for how industrial robots can better serve people in the new century, ensuring that people can share in the achievements of economic development.

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai conjectured that if $G$ is connected, then $p(G)\leq \lceil\frac{n}{2}\rceil$. In this paper, we prove that Gallai's path decomposition conjecture holds for the cartesian product $G\Box H$, where $H$ is any graph and $G$ is a unicyclic graph or a bicyclic graph.

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai conjectured that if $G$ is connected, then $p(G)\leq \lceil\frac{n}{2}\rceil$. Let $n_o(G)$ to denote the number of vertices with odd degree in $G$. Lov\'{a}sz proved that if $G$ is a connected graph with all vertices having degree odd, i.e. $n_o(G)=n$, then $p(G)=\frac n 2$. In this paper, we prove that if $G$ is a connected graph of order $m\geq 2$ with $p(G)=\frac{n_o(G)}{2}$ and $H$ is a connected graph of order $n$, then $p(G\Box H)\leq\frac{mn}{2}$. Furthermore, we prove that $p(G)=\frac{n_o(G)}{2}$, if one of the following is hold: (\romannumeral1) $G$ is a tree; (\romannumeral2) $G=P_n\Box T$, where $n\geq 4$ and $T$ is a tree; (\romannumeral3) $G=P_n\Box H$, where $H$ is an even graph.