The density of $k$-cacti via excluding minors

Abstract: A \emph{$k$-cactus} generalizes forests and cacti by allowing each edge to lie on at most $k$ cycles. The maximum number of edges is classical for forests and cacti, but for $k$-cacti was known only for $k\le 4$. In this note we treat general $k$. The key idea is that bounding the cycles through each edge forces a $k$-cactus to exclude a large complete minor; in particular, the class of $k$-cacti is minor-closed. From this we prove that every $n$-vertex $k$-cactus has $O!\left(\frac{\log k}{\sqrt{\log\log k}}\,n\right)$ edges for all sufficiently large $k$, and a construction shows this is optimal up to a factor of $\sqrt{\log\log k}$.

• The paper proves that every n-vertex k-cactus excludes a complete minor of order about log k/log log k, enabling the Kostochka–Thomason theorem to give the upper bound O((log k/√log log k)n).
• It establishes that k-cacti are minor-closed by lifting cycles through minor models and counts the cycles containing an edge in K_r using the expression ⌊e(r−2)!⌋−1.
• A construction formed by coalescing complete graphs gives infinitely many n with Ω((log k/log log k)n) edges, leaving only a √log log k gap and motivating the conjectured tight bound O((log k/log log k)n).

Overview

A $k$-cactus is a graph in which each edge lies on at most $k$ cycles, generalizing forests ($k=0$) and cacti ($k=1$). Zhang and Huang previously determined the maximum size of an $n$-vertex $k$-cactus for $k \le 4$ via a block-decomposition case analysis whose complexity grows rapidly with $k$, leaving the problem open for all $k \ge 5$ (2606.06298). This paper resolves the general-$k$ regime asymptotically by a different route: the local cycle condition forces exclusion of a large complete minor, and the Kostochka–Thomason extremal function for complete minors then bounds the edge count. The main result is that every $k$0-vertex $k$1-cactus has

$k$2

with a matching construction showing this is optimal up to a factor of $k$3.

A benchmark bound from block decomposition

As a warm-up, the authors derive a $k$4-dependent bound using only their earlier result that a 2-connected $k$5-cactus on $k$6 vertices has at most $k$7 edges. Each block $k$8 with $k$9 vertices satisfies $k=0$0, and a secant-line argument on the convex function $k=0$1 shows both terms are bounded by $k=0$2, where $k=0$3 is the root of $k=0$4. Summing over blocks yields:

$k=0$5

a coefficient of order $k=0$6. The minor-based argument below improves the dependence on $k=0$7 from polynomial to polylogarithmic.

Minor-closedness of $k=0$8-cacti

The structural heart of the paper is a cycle-lifting proposition: if $k=0$9 is a minor of $k=1$0, then for every edge $k=1$1 there is an edge $k=1$2 with $k=1$3. The proof fixes representative edges of an $k=1$4-model and lifts each cycle of $k=1$5 through $k=1$6 to a distinct cycle of $k=1$7 through $k=1$8, using connectedness of branch sets and disjointness to guarantee injectivity. Consequently, the class of $k=1$9-cacti is minor-closed for every fixed $n$0, extending the classical facts that forests are exactly the $n$1-minor-free graphs and cacti are exactly the diamond-minor-free graphs.

The quantitative version uses the exact count of cycles through a fixed edge of $n$2: deleting the edge from such a cycle leaves a nontrivial $n$3–$n$4 path, and counting paths with $n$5 internal vertices gives $n$6, proved via the Taylor expansion of $n$7. Hence any graph containing a $n$8 minor has some edge on at least $n$9 cycles, so if this quantity exceeds $k$0, no $k$1-cactus contains a $k$2 minor. Since this number grows super-exponentially in $k$3, the excluded complete minor is large relative to $k$4.

Upper bound via Kostochka–Thomason

Let $k$5. Every $k$6-vertex $k$7-cactus excludes $k$8 as a minor, so the Kostochka–Thomason theorem (every graph with more than $k$9 edges contains a $k \le 4$0 minor, where $k \le 4$1 is absolute) gives $k \le 4$2. Estimating $k \le 4$3 reduces to solving $k \le 4$4, i.e., $k \le 4$5; the authors show $k \le 4$6 for sufficiently large $k \le 4$7, using only the elementary inequality $k \le 4$8. Substituting yields the main corollary, $k \le 4$9, and a conservative explicit constant gives $k$0.

Lower-bound construction

For any $k$1 with $k$2, the graph $k$3 is itself a $k$4-cactus, and coalescing $k$5 copies at a common vertex preserves the property (every cycle lies within one block). This produces, for infinitely many $k$6, $k$7-cacti with exactly $k$8 edges. Choosing $k$9 — admissible since then $k \ge 5$0 — yields infinitely many $k \ge 5$1-vertex $k \ge 5$2-cacti with

$k \ge 5$3

Thus the upper and lower bounds differ only by a factor of $k \ge 5$4, and the maximum size of an $k \ge 5$5-vertex $k \ge 5$6-cactus is determined up to that power. The authors conjecture the lower bound is tight.

Limitations and open questions

Several caveats qualify the results. The upper bound holds only for sufficiently large $k \ge 5$7, inherits the unoptimized absolute constant from the Kostochka–Thomason theorem, and loses a $k \ge 5$8 factor against the construction; closing this gap is posed explicitly as the open problem of whether every $k \ge 5$9-vertex $k$0-cactus satisfies $k$1. Additionally, the lower-bound construction applies only to infinitely many values of $k$2 rather than all $k$3, and the block-decomposition route offers no improvement over $k$4, so any tightening must come from finer use of the excluded-minor structure or from direct arguments exploiting the cycle condition beyond mere minor exclusion.

Conclusion

This paper extends the extremal theory of $k$5-cacti from $k$6 to arbitrary $k$7 by establishing that the class is minor-closed and applying the Kostochka–Thomason theorem. The resulting bounds, $k$8 from above and $k$9 from below, reduce the problem to determining the correct power of $k$00, which the authors conjecture should vanish entirely.