Given a simple graph $G$, its line graph, denoted by $L(G)$, is obtained by representing each edge of $G$ as a vertex, with two vertices in $L(G)$ adjacent whenever the corresponding edges in $G$ share a common endpoint. By applying the line graph operation repeatedly, we obtain higher order line graphs, denoted by $L^{r}(G)$. In other words, $L^{0}(G) = G$, and for any integer $r \ge 1$, $L^{r}(G) = L(L^{r-1}(G))$. Given a graph $G$ on $n$ vertices, we wish to efficiently find out (i) if $L^k(G)$ has an Euler path, (ii) the value of $Δ(L^k(G))$. Note that the size of a higher order line graph could be much larger than that of $G$. For the first question, we show that for a graph $G$ with $n$ vertices and $m$ edges the largest $k$ where $L^k(G)$ has an Euler path satisfies $k = \mathcal O(nm)$. We also design an $\mathcal{O}(n^2m)$-time algorithm to output all $k$ such that $L^k(G)$ has an Euler path. For the second question, we study the growth of maximum degree of $L^k(G)$, $k \ge 0$. It is easy to calculate $Δ(L^k(G))$ when $G$ is a path, cycle or a claw. Any other connected graph is called a prolific graph and we denote the set of all prolific graphs by $\mathcal G$. We extend the works of Hartke and Higgins to show that for any prolific graph $G$, there exists a constant rational number $dgc(G)$ and an integer $k_0$ such that for all $k \ge k_0$, $Δ(L^k(G)) = dgc(G) \cdot 2^{k-4} + 2$. We show that $\{dgc(G) \mid G \in \mathcal G\}$ has first, second, third, fourth and fifth minimums, namely, $c_1 = 3$, $c_2 = 4$, $c_3 = 5.5$, $c_4 = 6$ and $c_5=7$; the third minimum stands out surprisingly from the other four. Moreover, for $i \in \{1, 2, 3, 4\}$, we provide a complete characterization of $\mathcal G_i = \{dgc(G) = c_i \mid G \in \mathcal G \}$. Apart from this, we show that the set $\{dgc(G) \mid G \in \mathcal G, 7 < dgc(G) < 8\}$ is countably infinite.

翻译：给定一个简单图$G$，其线图记作$L(G)$，通过将$G$的每条边表示为一个顶点来构造，当$G$中对应边共享一个公共端点时，$L(G)$中的两个顶点相邻。通过重复应用线图操作，我们得到高阶线图，记作$L^{r}(G)$。换言之，$L^{0}(G) = G$，且对于任意整数$r \ge 1$，$L^{r}(G) = L(L^{r-1}(G))$。给定一个具有$n$个顶点的图$G$，我们希望高效地确定：(i) $L^k(G)$是否具有欧拉路径；(ii) $Δ(L^k(G))$的值。注意高阶线图的规模可能远大于$G$。对于第一个问题，我们证明对于具有$n$个顶点和$m$条边的图$G$，使得$L^k(G)$具有欧拉路径的最大$k$满足$k = \mathcal O(nm)$。我们还设计了一个$\mathcal{O}(n^2m)$时间算法，输出所有使得$L^k(G)$具有欧拉路径的$k$值。对于第二个问题，我们研究$L^k(G)$（$k \ge 0$）的最大度增长。当$G$为路径、环或爪形图时，$Δ(L^k(G))$易于计算。任何其他连通图称为增殖图，所有增殖图的集合记作$\mathcal G$。我们扩展了Hartke和Higgins的工作，证明对于任意增殖图$G$，存在一个常数有理数$dgc(G)$和一个整数$k_0$，使得对所有$k \ge k_0$，有$Δ(L^k(G)) = dgc(G) \cdot 2^{k-4} + 2$。我们证明$\{dgc(G) \mid G \in \mathcal G\}$具有第一、第二、第三、第四和第五最小值，分别为$c_1 = 3$、$c_2 = 4$、$c_3 = 5.5$、$c_4 = 6$和$c_5=7$；其中第三最小值与其他四个值相比显著不同。此外，对于$i \in \{1, 2, 3, 4\}$，我们完整刻画了$\mathcal G_i = \{dgc(G) = c_i \mid G \in \mathcal G \}$的图类特征。除此之外，我们证明集合$\{dgc(G) \mid G \in \mathcal G, 7 < dgc(G) < 8\}$是可数无限的。