Determine the extremal normalized additive gap

Determine the extremal normalized additive gap c_Δ for connected graphs of maximum degree exactly Δ, or sharpen the existing lower and upper bounds on c_Δ.

Background

The paper defines c_Δ as the supremum, over connected graphs G with maximum degree exactly Δ, of the additive violation of the Henning–Yeo bound divided by the order of G. The constructed graph families provide explicit lower bounds, while a Caro–Wei argument applied to the square of G gives a general upper bound.

The authors prove only the asymptotic estimate c_Δ=Θ(1/Δ), with matching order of magnitude but nonmatching constants. They explicitly leave open the determination of the exact values of c_Δ and the improvement of the displayed lower and upper estimates.

References

We leave open whether the additive gap is unbounded at fixed degree four or five; from degree six onward, \cref{cor:all-fixed-degrees} gives an affirmative answer.

\begin{problem}\label{prob:corrected} Determine $c_\Delta$, or sharpen the lower and upper bounds on $c_\Delta$ in eq:general-c-lower and eq:c-upper. \end{problem}

eq:general-c-lower:

cΔ≥max⁡t,r≥2t+r=Δ(t−1)(r−1)(Δ2+1)(t+2r+2).c_\Delta\ge \max_{\substack{t,r\ge2\\t+r=\Delta}} \frac{(t-1)(r-1)} { (\Delta^2+1)(t+2r+2)}.

eq:c-upper:

cΔ≤(Δ+Δ−1−1)2Δ2+1.c_\Delta\le \frac{\bigl(\sqrt{\Delta+\Delta^{-1}}-1\bigr)^2} {\Delta^2+1}.

— Connected Counterexamples to the Henning--Yeo Conjecture on Identifying Vertex Covers  (2608.19455 - Wang, 19 Aug 2026) in Section 5, paragraph preceding Problem 5.1 and Problem 5.1 (labeled \cref{prob:corrected})

We leave open whether the additive gap is unbounded at fixed degree four or five; from degree six onward, \cref{cor:all-fixed-degrees} gives an affirmative answer.

— Connected Counterexamples to the Henning--Yeo Conjecture on Identifying Vertex Covers  (2608.19455 - Wang, 19 Aug 2026) in Section 5, paragraph immediately preceding Problem 5.1

He modified the conjecture to be

\lim_{n\to\infty}n2\bigl(\Delta-\lambda_1(n,\Delta)\bigr)= \begin{cases} (\Delta-1)\pi2/4,&\text{if $\Delta$ is odd};\ (\Delta-2)\pi2/2,&\text{if $\Delta$ is even}, \end{cases}

and proved the cases $\Delta\in{3,4}$.

— Extremal spectral radius of nonregular graphs with a fixed odd maximum degree  (2609.26044 - Huang et al., 22 Sep 2026) in Section 1, Introduction and main result