A local strengthening of Reeds {omega}, Delta, {chi} conjecture for quasi-line graphs

Reed’s $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi\leq \lceil\frac 12(\Delta+1+\omega)\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a colouring that achieves the bound of Reed’s original conjecture.

💡 Research Summary

The paper tackles a refined version of Reed’s famous ω‑Δ‑χ conjecture, which asserts that every graph G satisfies χ(G) ≤ ⌈½(Δ(G)+1+ω(G))⌉. While the original conjecture has been proved for all claw‑free graphs, the authors focus on a “local strengthening” proposed by King: for each vertex v, define ω(v) as the size of the largest clique containing v, and consider the bound max₍v∈V(G)⌉ ⌈½(d(v)+1+ω(v))⌉. The conjecture claims that χ(G) never exceeds this vertex‑wise maximum.

The authors first address line graphs, which are the intersection graphs of edges of a multigraph G. By translating the vertex‑coloring problem on a line graph L(G) into an edge‑coloring problem on G, they introduce three quantities: µ_G(uv), the multiplicity of the edge uv; t_G(uv), the maximum number of edges among triangles that contain uv; and the degrees d(u), d(v). They define a new parameter γ′ₗ(G) as the maximum over all edges uv of three expressions that combine these quantities. They prove that the chromatic index χ′(G) is bounded by γ′ₗ(G) and, crucially, that a γ′ₗ(G)‑edge‑coloring can be found in O(m²) time, where m is the number of edges of G.

The algorithmic heart of the proof lies in a generalized notion of Vizing’s fan. A fan is built around a single uncolored edge e and a partial coloring of the rest of the graph; it records a sequence of neighboring vertices whose incident colors interact in a controlled way. The authors develop a suite of lemmas (6‑11) that show how to extend a partial coloring when the fan has size two or at least three, how to enlarge a fan to a maximal one efficiently, and how to resolve conflicts by swapping colors along alternating paths. When the number of colors k is at least γ′ₗ(G), these lemmas guarantee that one can always complete the coloring in O(k + m) time.

Having established the local strengthening for line graphs, the authors invoke earlier structural results by Chudnovsky and Seymour that any quasi‑line graph can be decomposed into line‑graph‑like pieces. Because every quasi‑line graph is claw‑free and every vertex is bisimplicial (its neighborhood is the complement of a bipartite graph), the fan‑based technique extends without substantial modification. Consequently, they prove Theorem 4: for any quasi‑line graph G, χ(G) ≤ max₍v∈V(G)⌉ ⌈½(d(v)+1+ω(v))⌉.

On the algorithmic side, the line‑graph case yields an O(n²) time algorithm (n = number of vertices of the line graph), which improves upon the best known algorithm for achieving Reed’s original bound on line graphs. For quasi‑line graphs, the algorithm runs in O(n³ m²) time (n vertices, m edges), providing the first polynomial‑time method to construct a coloring meeting the locally strengthened bound for this class.

In summary, the paper delivers both a theoretical advance—proving King’s local strengthening of Reed’s conjecture for line graphs and, by extension, all quasi‑line graphs—and practical algorithms that construct optimal colorings within the new bound. The work blends structural graph theory (bisimplicial neighborhoods, claw‑free properties) with refined edge‑coloring techniques (generalized fans, color swaps), and it opens the door for further exploration of vertex‑local coloring bounds in broader graph families.