Factor-two bound for rich-neighbor chromatic index

Prove that every finite loopless multigraph G satisfies the rich-neighbor chromatic-index bound r(G)≤2χ′(G), improving the established universal bound r(G)≤3χ′(G).

Background

The paper establishes that every proper k-edge-coloring of a loopless multigraph can be refined into a rich-neighbor edge-coloring using at most 3k colors, and consequently that r(G)≤3χ′(G). The authors conjecture that the factor three can be reduced to two for all multigraphs.

The conjectured coefficient is asymptotically optimal: for complete bipartite graphs K_{Δ,Δ}, the paper proves r(K_{Δ,Δ})/χ′(K_{Δ,Δ})=2−1/Δ. The conjecture is verified in the paper for complete bipartite graphs and complete graphs, as well as in particular cases such as K_{2,2}, K_4, and K_6. The authors identify a possible route through total edge-dominating almost-star forests for which every monochromatic conflict graph is bipartite, but do not establish that such forests and colorings always exist.