Scope and significance of the Vizing phenomenon

Characterize the extent and theoretical significance of the phenomenon in which one graph invariant exceeds another related invariant by at most a small constant, as exemplified by bounds such as Vizing’s theorem and the surveyed relationships among maximum degree, matching book thickness, linear arboricity, book thickness, and star arboricity book thickness.

Background

The paper observes that several pairs of graph invariants exhibit behavior analogous to the pair (ε, μ), where the second invariant differs from the first by at most a small amount. Examples listed include maximum degree versus matching book thickness, ⌈Δ/2⌉ versus linear arboricity, and book thickness versus star arboricity book thickness.

The authors explicitly leave unresolved the general scope of this recurring pattern and its broader theoretical meaning.