Connectivity preserving spanning $(u,v)$-paths in $k$-connected graphs

Abstract: Hasunuma [Graphs Combin. 41:10 (2025)] proved that for $k\ge 2$, there exists a function $f(k)=O(k)$ such that every $k$-connected graph $G$ of order $n\ge f(k)$ with $δ(G)\ge \frac{n}{2}$ contains a Hamiltonian cycle $H$ such that $G-E(H)$ is $k$-connected. In this paper, we show that for $k\ge 2$, if $G$ is a $k$-connected graph of order $n\ge 6k+6$ with minimum degree at least $\frac{n+1}{2}$, then for any two distinct vertices $u,v\in V(G)$, there exists a Hamiltonian $(u,v)$-path $P$ such that $G-E(P)$ is $k$-connected. Moreover, we further extend this result to $s$ internally disjoint spanning $(u,v)$-paths.

• The paper establishes that in k-connected graphs with order n ≥ 6k+6 and minimum degree at least ⎡(n+1)/2⎤, a Hamiltonian (u,v)-path exists such that removing its edges preserves k-connectivity.
• It further demonstrates that for t ≥ 3, under stricter minimum degree conditions, one can construct s internally disjoint spanning (u,v)-paths while retaining k-connectivity post edge removal.
• The work employs Menger's theorem and careful counting arguments to optimize degree thresholds, offering valuable insights for robust network design and fault tolerance.

Connectivity Preserving Spanning $(u,v)$-Paths in $k$-Connected Graphs

Background and Motivation

Graph connectivity and Hamiltonian structures constitute central topics in graph theory, particularly in the context of $k$-connected graphs and the preservation of connectivity after the removal of certain spanning substructures. The paper addresses variants of classical conjectures and results that bound minimum degree conditions to guarantee the existence of subgraphs (paths, cycles, trees) whose removal leaves the host graph sufficiently connected.

Previous work established fundamental degree thresholds for the existence of Hamiltonian cycles or paths that, when removed, preserve $k$-connectivity. Notably, Hasunuma proved that for sufficiently large $k$-connected graphs with minimum degree at least $\frac{n}{2}$, there exists a Hamiltonian cycle whose edge removal does not decrease connectivity below $k$. This paper extends the inquiry to endpoint-prescribed Hamiltonian paths and systems of internally disjoint spanning $(u,v)$-paths, both under stricter degree conditions.

Main Theorems and Contributions

Two principal results are established:

1. Hamiltonian Path Connectivity Preservation:

For $k\geq 2$, if $G$ is a $k$0-connected graph of order $k$1 with minimum degree $k$2, then for any two distinct vertices $k$3, there exists a Hamiltonian $k$4-path $k$5 such that $k$6 remains $k$7-connected. This result sharpens the structure-preserving guarantee from cycles to endpoint-prescribed paths, under a higher minimum degree threshold than previous cycle-based results.

2. Internally Disjoint Spanning $k$8-Path Systems:

For $k$9 and $k$0, given a $k$1-connected graph of order $k$2 with minimum degree depending on $k$3, namely $k$4 for $k$5 and $k$6 for $k$7, it is shown that for any distinct $k$8 and any $k$9, there exist $k$0 internally vertex-disjoint spanning $k$1-paths $k$2 such that $k$3 is $k$4-connected.

These theorems extend the notion of spanning connectivity ($k$5), coupling classical Hamiltonicity and multipath systems to achieve edge-removal tolerance without connectivity degradation.

Techniques and Proof Structure

The proofs utilize established results such as Menger's theorem, degree-based sufficient conditions for Hamiltonian connectivity, and spanning connectivity lower bounds due to Lin-Huang-Hsu. A combination of arguments shows that:

1. Starting from a Hamiltonian $k$6-path or a spanning $k$7-path system, if $k$8 is not $k$9-connected, a separation procedure identifies a cutset causing disconnected components.
2. Degree constraints ensure that these components have sufficient size, limiting the number of components to two, and guaranteeing, via Menger-type arguments and degree counting, that internal vertices of critical paths (used in bridging components) have adequate neighborhood connections to restore $k$0-connectivity.
3. The process constructs auxiliary $k$1-connected subgraphs, linking them with vertex-disjoint paths whose internal vertices are contained in the minimal separator, then recovers the complete $k$2-connected structure using vertex addition lemmas.

Numerical thresholds are determined with careful counting arguments, bounding the sizes of components and separators, and optimizing the degree conditions required for the method's effectiveness.

Numerical Strength and Contradictory Claims

The paper asserts that under its degree conditions, Hamiltonian spanning path structures (not just cycles) can be removed while preserving $k$3-connectivity, a stricter regime than prior work and one that demonstrates convincingly strong numerical thresholds. Specifically, the minimum degree requirement for the existence of a Hamiltonian $k$4-path whose removal leaves $k$5-connectivity ($k$6) is higher than the analogous requirement for cycles. This distinguishes the result from Hasunuma's earlier Hamiltonian cycle guarantee and demonstrates uncompromising preservation of endpoint connectivity. The extension to $k$7 internally disjoint spanning $k$8-paths is also notable, providing resilient multipath structures.

Implications and Future Avenues

These results have theoretical implications for extremal structure-preserving graph design, network fault tolerance, and robust communication systems, where guaranteed path redundancy and connectivity preservation are critical. The formalism advances the known degree thresholds for prescribed spanning path systems, suggesting possible avenues for minimum degree reductions and generalizations to trees or other spanning structures.

A future direction, posed explicitly by the authors, is whether the minimum degree can be reduced to $k$9 for all $\frac{n}{2}$0, which would further align spanning path results with Dirac-type conditions and possibly illuminate a deeper relationship between spanning connectivity and classical Hamiltonicity.

Conclusion

The paper presents authoritative results on connectivity-preserving Hamiltonian and spanning path structures in $\frac{n}{2}$1-connected graphs, utilizing rigorous combinatorial arguments to establish strong degree conditions for the existence of endpoint-prescribed Hamiltonian paths and systems of disjoint spanning $\frac{n}{2}$2-paths whose removal does not compromise $\frac{n}{2}$3-connectivity. The results sharpen and extend previous theorems, set new numerical benchmarks, and open a pathway for further investigations into degree thresholds and structure-preserving subgraph systems.