Counterexamples to the Henning--Yeo Conjecture: Unbounded Fixed-Degree Gaps and Sharp First-Order Asymptotics

Abstract

Henning and Yeo conjectured an upper bound on the identifying vertex cover number of a graph in terms of its order, size, and maximum degree. A two-parameter family $H_{t,r}$ of connected diameter-two graphs disproves the bound for every maximum degree at least four; after denominators are cleared, its margin is exactly $-(t-1)(r-1)$. The complement relation $τ_D=n-ρ$ exposes the mechanism: diameter-two fibres admit at most one packing vertex, while degree deficit accumulates under tree gluing with controlled port loads. Writing $A_Δ$ for the supremal additive gap at maximum degree exactly $Δ$, an exact transfer formula gives $A_Δ=+\infty$ for every $Δ\ge 4$, using Petersen fibres in degrees four and five and the original $H_{t,r}$ blocks in higher degrees. If $c_Δ$ denotes the corresponding supremal gap per vertex, rooted rook-graph fibres match a universal square-graph packing bound to first order. Consequently, $c_Δ\sim 1/Δ$, equivalently $Δc_Δ\to 1$ as $Δ\to\infty$.