关于识别顶点覆盖的Henning-Yeo猜想的连通反例
Counterexamples to the Henning--Yeo Conjecture: Unbounded Fixed-Degree Gaps and Sharp First-Order Asymptotics
AI总结：
该研究构造双参数图族作为反例，证明Henning-Yeo关于识别顶点覆盖数的猜想不成立，且最大度≥6时加性差距无界，最小反例为8阶图$H_{2,2}$。
AI中文摘要：
Henning和Yeo提出了一个图的识别顶点覆盖数关于其阶数、边数和最大度的上界猜想。我们通过一个双参数连通直径为2的图族$H_{t,r}$证明该猜想的不等式不成立。通分后，右侧减左侧恰好为$-(t-1)(r-1)$，因此对于每个最大度至少为4的情况，都存在连通反例。通过低度数顶点连接图的副本可保持最大度，并能精确确定填充数。在最大度为5时，这给出了阶数任意大、加性差距为1/13的反例；对于每个固定的最大度$\triangle\boldsymbol{\text{≥}}6$，合适的链具有无界的加性违反。因此，无论是取整还是固定加性修正都无法修复该猜想。最大度为$\triangle$时的最优归一化加性差距为$\boldsymbol{\text{Θ}}(1/\triangle)$。对所有阶数不超过7的图进行穷举检查显示，8阶示例$H_{2,2}$具有最小可能的阶数。
英文摘要：
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 $Δ\to\infty$.
