On Euler Paths and the Maximum Degree Growth of Iterated Higher Order Line Graphs

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.

💡 Research Summary

This paper investigates two fundamental algorithmic questions concerning iterated line graphs of a simple connected graph G: (i) determining for which iteration k the graph L⁽ᵏ⁾(G) possesses an Euler path, and (ii) computing the maximum degree Δ(L⁽ᵏ⁾(G)). The authors introduce a novel structural tool—the notion of a “critical edge,” an edge whose incident vertices have opposite parity degrees. They prove that vertices of odd degree in L(G) correspond bijectively to critical edges of G (Observation 11). Using this correspondence, they perform a exhaustive case analysis of how critical edges can appear in L(G) and L²(G). From these local configurations they derive global constraints on G that dictate whether L⁽ᵏ⁾(G) can have an Euler path. Their main combinatorial result is that the largest k for which an Euler path can exist is bounded by O(n·m), where n and m are the numbers of vertices and edges of the original graph. Moreover, they present an explicit algorithm that, in O(n²·m) time, enumerates all such k values. This algorithm avoids the exponential blow‑up that would result from naïvely constructing each iterated line graph.

The second part of the work extends the Maximum Degree Growth Property (MDGP) originally studied by Hartke and Higgins. The authors focus on “prolific graphs,” defined as all connected graphs except paths, cycles, and the claw K₁,₃. For any prolific graph G they prove the existence of a rational constant dgc(G) and an integer k₀ such that for all k ≥ k₀, \