Hitting Maximum Independent Sets in Dense and Highly Connected Graphs

Abstract: For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollobás--Erdős--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate [ h(G)\le \left\lfloor\frac{|V(G)|}{2α(G)+δ(G)-|V(G)|}\right\rfloor ] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $κ(G)\geρn$ and $ρ&gt;1/3$ has a hitting set of size at most $\lfloor(ρ-1/3)<sup>{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $κ(G)&gt;4n/9$ and to the sharp bound $3$ when $κ(G)&gt;n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=Ω(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.

• The paper shows that proving sublinear hitting numbers for maximum independent sets is equivalent to doing so in dense regular graphs and, within hereditary classes, graphs with any fixed positive linear connectivity.
• The paper establishes the sharp bound h(G) ≤ ⌊n/(2α(G)+δ(G)−n)⌋ when 2α(G)+δ(G)>n, yielding constant transversals for highly connected 3-colorable graphs and the optimal bound h(G)≤3 above connectivity n/2.
• The paper combines container and coloring methods to obtain O(log n) bounds in several dense or near-regular regimes, while switching and shift-graph constructions show that Ω(√n) hitting numbers remain possible across broad parameter ranges.

The parameter and its context

For a graph $G$, let $M(G)$ denote the family of maximum independent sets, and let $h(G)$ be the minimum size of a vertex set meeting every member of $M(G)$. Equivalently, $h(G)$ is the least number of vertices whose deletion lowers the independence number, so the parameter is simultaneously a transversal problem for an extremal set system and a stability question for $\alpha(G)$. The driving open problem is the Bollobás–Erdős–Tuza conjecture from 1991: for every fixed $c\in(0,1]$, every $n$-vertex graph with $\alpha(G)\ge cn$ satisfies $h(G)=o(n)$. Hajnal's intersection–union inequality gives $M(G)$0 when $M(G)$1, but no general sublinear bound is known below that threshold, and Alon's shift-graph construction shows that linear independence number alone permits $M(G)$2 (Alon, 2021). The paper under review, by Bai, Chang, and Yan, contributes two structural reduction principles showing that density and connectivity do not simplify the conjecture, together with several sharp quantitative bounds in dense regular and highly connected 3-colorable settings.

Two reductions: regularity and connectivity preserve full difficulty

The first main result is an equivalence rather than a one-way implication. Writing $M(G)$3 for the maximum of $M(G)$4 over $M(G)$5-vertex graphs with $M(G)$6, and $M(G)$7 for the same maximum restricted to regular graphs of degree at least $M(G)$8, the authors prove that $M(G)$9 for all fixed $h(G)$0 if and only if $h(G)$1 for all fixed $h(G)$2, for any single fixed $h(G)$3. The forward direction is trivial; the converse uses an augmented switching graph construction. Given any graph $h(G)$4 on $h(G)$5 vertices, one forms three disjoint copies and applies the switching operation (two layers with same-layer edges mirroring adjacency and crossed edges mirroring non-adjacency), then adds antipodal matching edges. The result is $h(G)$6-regular on $h(G)$7 vertices with $h(G)$8 equal to $h(G)$9 and $M(G)$0 exactly $M(G)$1; a complete join of $M(G)$2 copies then tunes the degree ratio to $M(G)$3 while preserving both parameters up to fixed factors. Consequently, imposing regularity together with any prescribed positive linear minimum degree does not weaken the conjecture: dense regular graphs already encode its full asymptotic difficulty.

The second reduction addresses connectivity within hereditary classes. A localization argument recursively deletes small vertex cuts, always passing to a smallest component, and shows that each graph $M(G)$4 contains an induced subgraph $M(G)$5 with either $M(G)$6 or $M(G)$7 such that

$M(G)$8

Since hereditary classes are closed under taking induced subgraphs, this yields the equivalence: within any hereditary class $M(G)$9, a uniform sublinear bound $h(G)$0 holds if and only if it holds at every fixed positive connectivity ratio. In particular, resolving the conjecture for highly connected 3-colorable graphs—for every positive connectivity ratio—would resolve it for all 3-colorable graphs. The authors note plainly that the quantifier "for every fixed $h(G)$1" is essential: a bound at a single fixed $h(G)$2 leaves a linear separator cost.

A sharp general upper bound via degree surplus

The quantitative core of the paper is a bound requiring neither regularity nor chromatic assumptions. If $h(G)$3, then

$h(G)$4

and this estimate is sharp for every balanced complete multipartite graph. The proof partitions vertices by the family $h(G)$5 of maximum independent sets containing them, shows via Hajnal's inequality that each maximal equivalence class has size at least $h(G)$6, and selects one representative per maximal class. Applied to 3-colorable graphs, where $h(G)$7 and $h(G)$8, this immediately gives $h(G)$9 whenever $\alpha(G)$0 with $\alpha(G)$1: a constant transversal throughout the entire supercritical range. More generally, $\alpha(G)$2 implies $\alpha(G)$3. At the boundary $\alpha(G)$4, the proposition also forces any hypothetical sequence with linear hitting number to satisfy $\alpha(G)$5—extremely tight against both thresholds simultaneously.

Constant bounds from direct coloring arguments

Working directly with a 3-coloring improves the constants at higher connectivity. The key counting step shows that if a maximum independent set meets all three color classes, then $\alpha(G)$6. Hence:

• Connectivity above $\alpha(G)$7: if $\alpha(G)$8, no maximum independent set meets all three color classes; combining bipartite transversals (each worth at most 2, via König's theorem) over the three color-class pairs gives $\alpha(G)$9.
• Connectivity above $c\in(0,1]$0: a refined count rules out maximum independent sets meeting two color classes as well, so every maximum independent set is a color class, giving the sharp bound $c\in(0,1]$1.

The bound 3 is attained by the balanced complete tripartite graph $c\in(0,1]$2, which has connectivity $c\in(0,1]$3. The remark also observes that for 3-colorable $c\in(0,1]$4 with $c\in(0,1]$5, connectivity never exceeds $c\in(0,1]$6, so ratios above $c\in(0,1]$7 constitute no nontrivial regime.

Dense regular graphs: logarithmic bounds and square-root obstructions

For $c\in(0,1]$8-regular graphs with $c\in(0,1]$9 and $n$0, the paper obtains a layered set of results. When $n$1, a container argument (Alon's regular-graph container lemma combined with Hajnal cores and random sampling) yields $n$2, strictly improving Alon's $n$3 estimate in the linear-degree regime. When $n$4, the degree-surplus bound becomes constant: $n$5. When $n$6, a cardinality refinement of King's maximum-clique transversal theorem applied to the complement gives $n$7.

These upper bounds are not uniform across the parameter plane. Joining $n$8 copies of Alon's shift graph produces regular graphs with degree ratio exceeding $n$9, independence ratio above $\alpha(G)\ge cn$0, and hitting number $\alpha(G)\ge cn$1; the augmented switching operation yields $\alpha(G)\ge cn$2-regular examples with $\alpha(G)\ge cn$3 and $\alpha(G)\ge cn$4. Thus for every $\alpha(G)\ge cn$5 there exist regular graphs of linear degree and linear independence number with $\alpha(G)\ge cn$6: linear degree alone cannot force logarithmic transversals over the full feasible range ($\alpha(G)\ge cn$7, $\alpha(G)\ge cn$8).

Near-regular 3-colorable graphs and the critical family at $\alpha(G)\ge cn$9

Below the one-third connectivity threshold, the paper offers a different route under balanced degrees: for fixed $h(G)=o(n)$0, every sufficiently large 3-colorable graph with $h(G)=o(n)$1 and $h(G)=o(n)$2 satisfies $h(G)=o(n)$3. The proof runs a greedy fingerprinting algorithm recording high-degree vertices of an unknown independent set, groups maximum independent sets by their fingerprints into at most $h(G)=o(n)$4 containers of size at most $h(G)=o(n)$5, and samples $h(G)=o(n)$6 vertices to hit all common cores.

A critical construction explains why these methods stop at the boundary. The graph $h(G)=o(n)$7 consists of three independent sets $h(G)=o(n)$8 of size $h(G)=o(n)$9, with $M(G)$00 joined completely to $M(G)$01 and a perfect matching between $M(G)$02 and $M(G)$03. It is 3-colorable with $M(G)$04, $M(G)$05, and $M(G)$06—so it is not a counterexample—but it has exponentially many maximum independent sets and VC dimension at least $M(G)$07. Its degree surplus $M(G)$08 equals exactly $M(G)$09, while $M(G)$10. Consequently, the sharp degree-surplus bound degenerates to the trivial estimate and the near-regular hypothesis fails: the family simultaneously marks the limits of both arguments without itself obstructing the conjecture.

Limitations and open problems

The paper is candid about where its techniques end. In the dense regular setting, the container argument requires $M(G)$11, and the square-root constructions show that no uniform $M(G)$12 bound can hold across the whole region $M(G)$13. This leaves a concrete gap: determine the asymptotic order of $M(G)$14 in that region—in particular, whether $M(G)$15 forces a constant bound depending only on $M(G)$16 and $M(G)$17, or whether $M(G)$18 holds throughout. In the 3-colorable setting, Corollary on the equivalence shows that a sublinear bound at every fixed positive connectivity ratio would settle the full conjecture for 3-colorable graphs, yet the present methods cover only $M(G)$19 (constantly) and near-regular instances below. Whether $M(G)$20 holds for every fixed $M(G)$21 remains open, and the critical family indicates that any progress must exploit structure beyond degree surplus and degree-ratio balance.

Conclusion

This paper reframes the Bollobás–Erdős–Tuza conjecture as equally hard on dense regular graphs and, within hereditary classes, on graphs of arbitrary fixed positive linear connectivity. Its sharp degree-surplus bound converts supercritical connectivity in 3-colorable graphs into constant transversals, reaching the optimal value 3 above $M(G)$22, while container and coloring arguments yield logarithmic bounds for dense regular graphs above independence ratio $M(G)$23 and for near-regular 3-colorable graphs. The matching square-root lower bounds and the critical family at connectivity $M(G)$24 delineate precisely where current techniques fail, reducing the remaining difficulty to two well-defined boundary regimes.