关于欧拉路径和迭代高阶线图的最大度增长
On Euler Paths and the Maximum Degree Growth of Iterated Higher Order Line Graphs
AI总结：
研究了高阶线图的欧拉路径存在性及最大度增长，提出了一种高效算法并确定了最大度的渐进行为。
摘要：
给定一个简单图G，其线图L(G)是通过将G中的每条边表示为一个顶点来构造的，当且仅当对应的边在G中共享一个公共端点时，L(G)中的两个顶点才相邻。通过反复应用线图操作，我们得到更高阶的线图，记为L^r(G)。换句话说，L^0(G)=G，对于任何整数r≥1，L^r(G)=L(L^{r-1}(G))。给定一个有n个顶点的图G，我们希望高效地确定（i）L^k(G)是否存在欧拉路径，（ii）Δ(L^k(G))的值。注意，更高阶的线图的规模可能远大于G。对于第一个问题，我们证明对于一个有n个顶点和m条边的图G，最大的k使得L^k(G)存在欧拉路径满足k=O(nm)。我们还设计了一个O(n²m)时间的算法，以输出所有满足L^k(G)存在欧拉路径的k。对于第二个问题，我们研究了L^k(G)的最大度的增长，k≥0。当G是路径、环或爪时，计算Δ(L^k(G))相对容易。任何其他连通图都被称为 prolific 图，并将所有 prolific 图的集合记为G。我们扩展Hartke和Higgins的工作，证明对于任何 prolific 图G，存在一个常数有理数dgc(G)和一个整数k₀，使得对于所有k≥k₀，Δ(L^k(G))=dgc(G)·2^{k-4}+2。我们证明{dgc(G) | G∈G}有第一、第二、第三、第四和第五个最小值，即c₁=3，c₂=4，c₃=5.5，c₄=6和c₅=7；第三小值意外地从其他四个中脱颖而出。此外，对于i∈{1,2,3,4}，我们对G_i={dgc(G)=c_i | G∈G}进行了完整刻画。此外，我们证明集合{dgc(G) | G∈G,7 < dgc(G) <8}是可数无限的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
