Closing paths to cycles in symmetric graphs

TL;DR

The paper investigates when paths close to cycles in symmetric graphs, formulating the PtoC and IndPtoC problems for vertex-transitive and edge-transitive classes. It provides complete resolutions for connected VT graphs, ET graphs, and ET nonstar graphs, showing that short paths inevitably close to cycles while longer paths admit infinite families of counterexamples; the induced-path variants mirror these patterns with similar finite thresholds and infinite counterexamples. Core techniques rely on automorphism-based symmetry properties, line graph relations, and connectivity bounds, yielding tight thresholds (notably length $4$ for VT, length $3$ for ET nonstars, and length $2$ for induced VT paths) and explicit counterexamples such as circulant graphs $\text{Circ}(2n,\{\pm1,\pm2\})$, the diamond graphs $K_n^{\diamond}$, and hypercube line graphs $L(Q_n)$. The results illuminate the precise extent symmetry enforces path-cycle closures and identify natural obstructions like star graphs, while pointing to rich avenues for future work in parametrized regularity and broader symmetry classes.

Abstract

It was shown by Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius in 2022 that every induced $2$-edge path in a vertex-transitive graph closes to an induced cycle. Similar results were obtained for 3-edge paths closing to cycles in edge-transitive graphs, where the cycle can be assumed to be induced if the path is induced. Motivated by these results, we consider the following problem: For a given class of graphs, determine all integers $\ell\geq 0$ such that for every graph in the class, every path of length at most $\ell$ closes to a cycle. We also consider the variant of the problem for induced paths closing to induced cycles. We completely solve these problems for the classes of (finite) vertex-transitive graphs, edge-transitive graphs, and edge-transitive graphs that are not stars. For all but one case of a negative answer, we provide infinite families of connected counterexamples.

Figures (6)

• Figure 1: The graph $G_n$ and a $5$-edge path in it that does not close to a cycle
• Figure 2: $K_3^\diamond$ and $K_4^\diamond$ with their respective $4$-edge paths not closing to cycles
• Figure 3: $K_3^\diamond$ and $L(K_3^\diamond)$ and induced paths not closing to any induced cycle
• Figure 4: A $13$-vertex edge-transitive graph.
• Figure 5: A $4$-regular graph with $\kappa(G)=2$ and a path of length $4$ that does not close to a cycle.

Theorems & Definitions (43)

• Theorem 1.1: BCGMS2022
• Theorem 1.2: BCGMS2022
• Theorem 1.3: BCGMS2022
• Theorem 1.4: PtoC for VT graphs
• Theorem 1.5: Induced PtoC for VT graphs
• Theorem 1.6: PtoC for non-star ET graphs
• Theorem 1.7: Induced PtoC for non-star ET graphs
• Theorem 2.1: godsil2001algebraic
• Theorem 2.2: godsil2001algebraic
• Theorem 2.3: Watkins MR266804