Strong chromatic index of k-degenerate graphs

A {\em strong edge coloring} of a graph $G$ is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} $\chiup_{s}’(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. In this note, we improve a result by D{\k e}bski \etal [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show that the strong chromatic index of a $k$-degenerate graph $G$ is at most $(4k-2) \cdot \Delta(G) - 2k^{2} + 1$. As a direct consequence, the strong chromatic index of a $2$-degenerate graph $G$ is at most $6\Delta(G) - 7$, which improves the upper bound $10\Delta(G) - 10$ by Chang and Narayanan [Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2) 119–126]. For a special subclass of $2$-degenerate graphs, we obtain a better upper bound, namely if $G$ is a graph such that all of its $3^{+}$-vertices induce a forest, then $\chiup_{s}’(G) \leq 4 \Delta(G) -3$; as a corollary, every minimally $2$-connected graph $G$ has strong chromatic index at most $4 \Delta(G) - 3$. Moreover, all the results in this note are best possible in some sense.

💡 Research Summary

The paper studies the strong edge‑coloring problem, where a coloring of the edges of a graph must be proper and each color class must form an induced matching. The minimum number of colors required is called the strong chromatic index χ′ₛ(G). While the general conjecture of Erdős and Nešetřil predicts a quadratic bound (≈5/4 Δ²), much tighter bounds are known for sparse graphs. In particular, for k‑degenerate graphs (graphs in which every subgraph has a vertex of degree at most k) previous work gave the bound (4k‑1)·Δ‑k(2k+1)+1 (Dębski‑Grytczuk‑Śleszyńska‑Nowak) and later (4k‑2)·Δ‑2k²+k+1 (Yu).

The authors improve these results by a careful greedy decomposition of the edge set into star‑shaped blocks. Using Lemma 1 (originally due to Chang and Narayanan) they repeatedly select a vertex wᵢ whose incident edges to low‑degree neighbours (degree ≤ k) form a star Λᵢ. The construction guarantees that distinct blocks have distinct centers. The edges are then colored in reverse order (from the last block to the first) using a greedy algorithm that respects the strong‑coloring condition.

The key technical contribution is a refined counting argument for the number of already colored edges that lie in the closed neighbourhood N(wᵢ)∪N(vᵢ) of the currently processed edge wᵢvᵢ. By partitioning the neighbours of wᵢ into three sets (Xᵢ, Yᵢ, Zᵢ) according to whether they belong to earlier blocks or have degree ≤ k or > k, and similarly handling the neighbours of vᵢ, the authors show that at most (4k‑2)·Δ‑2k² colors can be forbidden at any step. Consequently at least one unused color remains, and the whole graph can be colored with at most (4k‑2)·Δ‑2k²+1 colors.

When k = 2 this yields the explicit bound χ′ₛ(G) ≤ 6Δ‑7 for all 2‑degenerate graphs, improving earlier bounds of 10Δ‑10 (Chang‑Narayanan) and 8Δ‑4 (Luo‑Yu). The bound is tight up to an additive constant, as the 5‑cycle attains χ′ₛ=5, matching the formula for Δ=2.

The paper further investigates a subclass of graphs in which all vertices of degree at least three induce a forest. Lemma 2 shows that such a graph contains a vertex w with many neighbours of degree at most two. Applying the same star‑decomposition technique leads to the stronger bound χ′ₛ(G) ≤ 4Δ‑3 for this class. Since minimally 2‑connected graphs have exactly this structural property (their 3⁺‑vertices form a forest), the same bound holds for all minimally 2‑connected graphs.

The authors also comment on earlier work concerning chordless graphs, pointing out a flaw in the proof of an 8Δ‑6 bound and noting that the correct bound for chordless (hence 2‑degenerate) graphs is 4Δ‑3, which follows from their results. Because the algorithm is purely greedy, the bounds automatically extend to the list‑version of strong edge‑coloring.

In summary, the paper delivers a unified and improved upper bound for the strong chromatic index of k‑degenerate graphs, namely (4k‑2)·Δ‑2k²+1, and derives sharper linear bounds for several important subclasses (2‑degenerate, forest‑induced 3⁺‑vertices, minimally 2‑connected). These results advance the state of knowledge on strong edge‑colorings of sparse graphs and bring the bounds significantly closer to the conjectured optimum for many practical graph families.