Vertex-distinguishing edge coloring of graphs

Let $k \ge 1$ be an integer and let $G$ be a nonempty simple graph. An \emph{edge-$k$-coloring} $φ$ of $G$ is an assignment of colors from ${1,\ldots,k}$ to the edges of $G$ such that no two adjacent edges receive the same color. For a vertex $v \in V(G)$, we write $φ(v)$ for the set of colors assigned to the edges incident with $v$. The coloring $φ$ is called \emph{vertex-distinguishing} if $φ(u) \ne φ(v)$ for every pair of distinct vertices $u,v \in V(G)$. A vertex-distinguishing edge-$k$-coloring exists if and only if $G$ has at most one isolated vertex and no isolated edge. The least integer $k$ for which such a coloring exists is called the \emph{vertex-distinguishing chromatic index} of $G$, denoted $χ’{vd}(G)$. In 1997, Burris and Schelp conjectured that for every graph $G$ with at most one isolated vertex and no isolated edge, $ k(G) ;\le; χ’{vd}(G) ;\le; k(G)+1$, where $k(G)$ is the natural lower bound required for a vertex-distinguishing coloring in $G$. In 2004, Balister, Kostochka, Li, and Schelp verified the conjecture for graphs $G$ satisfying $Δ(G) \ge \sqrt{2|V(G)|} + 4 $ and $δ(G) \ge 5$. For graphs that do not satisfy these conditions, the best known general upper bound on $χ’{vd}(G)$ remains $|V(G)| + 1$, established in 1999 by Bazgan, Harkat-Benhamdine, Li, and Woźniak. In this paper, we prove that $χ’{vd}(G) \le \floor{5.5k(G)+6.5}$, which represents a substantial improvement over the bound $|V(G)| + 1$ whenever $k(G) = o(|V(G)|)$. We further show that $χ’_{vd}(G) \le k(G) + 3$, for all $d$-regular graphs $G$ with $d \ge \log_2 |V(G)|\geq 8$.

💡 Research Summary

This paper presents significant advancements in the study of vertex-distinguishing edge colorings (VDEC) of graphs. A VDEC is an edge coloring where no two adjacent edges share a color, and the set of colors incident to each vertex is unique across all vertices. The minimum number of colors required for such a coloring of a graph G is its vertex-distinguishing chromatic index, denoted χ′_vd(G). A fundamental conjecture by Burris and Schelp (1997) states that for any graph G admitting a VDEC, χ′_vd(G) is either k(G) or k(G)+1, where k(G) is a combinatorial lower bound depending on the number of vertices of each degree. While this conjecture was verified for graphs with high maximum degree and minimum degree at least 5, the best general upper bound for all other graphs remained |V(G)| + 1 for over two decades.

The authors break new ground with two main theorems. First, Theorem 1.2 establishes a new general upper bound: χ′_vd(G) ≤ ⌊5.5k(G) + 6.5⌋ for any graph G. This represents a substantial improvement over the |V(G)| + 1 bound whenever the natural parameter k(G) grows slower than the number of vertices (i.e., k(G) = o(|V(G)|)). The proof employs a sophisticated decomposition strategy. It first extracts a specific linear forest F from G (via Lemma 2.3), whose components are paths of lengths 2, 3, or 4. The remaining subgraph H is then given an optimal edge-k(G)-coloring φ. A key result from prior work (Theorem 2.1) guarantees that such an optimal coloring is “semi-vertex-distinguishing,” meaning any color set appears on at most two vertices. The core of the proof involves strategically recoloring two sets of edges: first, a carefully chosen set of edges incident to potential conflict pairs in H are recolored with k new colors to resolve conflicts within H; second, the edges of the linear forest F are recolored using at most ⌊3.5k+1⌋ further new colors (via Lemma 2.2) to resolve all remaining conflicts, including those between vertices in F and those between F and H. The sum of the original k(G) colors and the new colors used in these two steps yields the bound.

Second, Theorem 1.3 provides a much stronger bound for a large class of regular graphs. It proves that for any d-regular graph G with d ≥ log₂|V(G)| ≥ 8, we have χ′_vd(G) ≤ k(G) + 3. This bound is remarkably close to the k(G)+1 conjectured by Burris and Schelp. The proof follows a similar decomposition into a linear forest F and a subgraph H. Leveraging the regularity and the high-degree condition, the authors show that an optimal edge-k(G)-coloring of H is again semi-vertex-distinguishing. They then apply a lemma from earlier work (Lemma 2.4) which allows the edges of the path forest F to be colored using only three additional colors in a way that distinguishes all vertex color sets. Combining these gives the k(G)+3 bound.

Overall, this paper makes a major contribution to graph coloring theory by introducing a novel proof technique based on graph decomposition and optimal colorings, and by dramatically improving the known upper bounds for both general graphs and dense regular graphs, bringing us closer to resolving the long-standing Burris-Schelp conjecture.