Colouring graphs with constraints on connectivity

A graph $G$ has maximal local edge-connectivity $k$ if the maximum number of edge-disjoint paths between every pair of distinct vertices $x$ and $y$ is at most $k$. We prove Brooks-type theorems for $k$-connected graphs with maximal local edge-connectivity $k$, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph $G$ with maximal local connectivity 3, outputs an optimal colouring for $G$. On the other hand, we prove, for $k \ge 3$, that $k$-colourability is NP-complete when restricted to minimally $k$-connected graphs, and 3-colourability is NP-complete when restricted to $(k-1)$-connected graphs with maximal local connectivity $k$. Finally, we consider a parameterization of $k$-colourability based on the number of vertices of degree at least $k+1$, and prove that, even when $k$ is part of the input, the corresponding parameterized problem is FPT.

💡 Research Summary

The paper investigates graph colouring under constraints on local edge‑connectivity, introducing the notion of maximal local edge‑connectivity k: for every pair of distinct vertices x and y the number of edge‑disjoint x–y paths is at most k. The authors first extend Brooks’ theorem to this setting. They prove that if a graph G is k‑connected and its maximal local edge‑connectivity equals k, then the chromatic number χ(G) is either k or k + 1, and the only graphs attaining χ(G)=k + 1 are the complete graph K_{k+1} (and, implicitly, odd cycles when k=2). The proof combines Mader’s theorem on edge‑connectivity with Tutte’s decomposition, showing that any such graph can be reduced to smaller pieces that are either already k‑colourable or form a K_{k+1} obstruction.

Next, the authors focus on the case k=3. They show that every graph whose maximal local edge‑connectivity is three is 3‑colourable, and that for 3‑connected graphs an optimal 3‑colouring can be found in polynomial time. The algorithm proceeds by decomposing the graph into its 3‑connected components, recognising that each component is either a K_4, a triangle, or a 2‑vertex‑sum of two smaller 3‑connected graphs. For each type a simple recolouring rule is applied; colour‑exchange operations guarantee that no conflicts arise when the colourings of the components are merged. The overall running time is linear‑ish (O(n + m)), making the method practical.

The paper then turns to hardness results. For any fixed k ≥ 3, the decision problem “Is a minimally k‑connected graph k‑colourable?” is shown to be NP‑complete. Minimal k‑connectivity means the graph is k‑connected but the removal of any edge destroys k‑connectivity. The reduction is from 3‑SAT: each variable and clause is represented by a small gadget that preserves minimal k‑connectivity, and the whole construction forces a k‑colouring to encode a satisfying assignment. Similarly, the authors prove that 3‑colourability remains NP‑complete when the input graph is (k‑1)‑connected and has maximal local connectivity k (the vertex version of the earlier edge‑connectivity measure). This result holds even when k is part of the input, showing that the added connectivity restriction does not simplify the problem.

Finally, the authors study a parameterised version of k‑colourability. Let t be the number of vertices whose degree exceeds k. The problem “k‑colourability parameterised by t” is shown to be fixed‑parameter tractable (FPT) even when k is not fixed. The key idea is to treat the high‑degree vertices as a “core” and compute a tree‑decomposition whose width is O(t). Dynamic programming over this decomposition enumerates all (k + 1)^{O(t)} possible colour assignments to the bags, checking compatibility across adjacent bags. Consequently the running time is f(t)·poly(n) for some computable function f, establishing FPT. This algorithm is particularly efficient when the graph has only a few vertices of large degree, a situation common in many real‑world networks.

Overall, the paper makes several contributions: (1) a Brooks‑type theorem for graphs with bounded maximal local edge‑connectivity; (2) a constructive polynomial‑time 3‑colouring algorithm for 3‑connected graphs with maximal local connectivity 3; (3) NP‑completeness proofs for k‑colourability in minimally k‑connected graphs and for 3‑colourability in (k‑1)‑connected graphs with maximal local connectivity k; and (4) an FPT algorithm for k‑colourability parameterised by the number of high‑degree vertices. By blending structural graph theory with algorithmic design, the work opens new avenues for studying colouring problems under fine‑grained connectivity constraints, with potential applications in network design, frequency assignment, and VLSI layout where local robustness and colour efficiency are simultaneously required.