Goldberg's density conjecture

Let GG be a finite, undirected, loopless multigraph. Write Δ(G)\Delta(G) for its maximum degree, χ(G)\chi'(G) for its chromatic index, and define its density by

Γ(G):=maxHG, H2E(H)H/2.\Gamma(G):=\max_{H\subseteq G,\ |H|\ge2}\left\lceil\frac{|E(H)|}{\lfloor |H|/2\rfloor}\right\rceil.

Goldberg's density conjecture. For any graph GG, if Γ(G)Δ(G)1\Gamma(G)\le\Delta(G)-1, then

χ(G)=Δ(G).\chi'(G)=\Delta(G).

The conjecture concerns when the maximum degree alone determines the chromatic index. The source notes that the Goldberg--Seymour conjecture has recently been confirmed, but does not state that this density conjecture itself has been resolved.

Sources & referencesView supporting material

Primary source

Guantao Chen, Yuying Ma, Yimo Su and Shengze Wang, “Average degrees of edge-Δ-critical multigraphs”, arXiv:2606.12271 (2026).

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