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\title{Highly connected counterexamples to the\\ multiplicative Merino--Welsh inequality}
% IMPORTANT FOR SUBMISSION: Fill in the author field with the human contributor(s)
% and ensure the contributor names match those associated with ORCID on Hexagon.
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\author{}
\date{20 October 2026}
\hypersetup{pdftitle={Highly connected counterexamples to the multiplicative Merino-Welsh inequality},
pdfsubject={Matroids; Tutte polynomial; three conjecture counterexamples},
pdfkeywords={matroid, Merino-Welsh inequality, simple, cosimple, truncation, Tutte polynomial}}
\begin{document}
\maketitle
\begin{abstract}
For every integer $k\ge3$, we construct $k$-connected matroids representable
over every infinite field, with girth $k$, arbitrarily large cogirth,
and Merino--Welsh quotient
$\Phi(M)=T_M(1,1)^2/(T_M(2,0)T_M(0,2))$ tending to infinity.
In particular, this disproves \emph{Conjecture~7.2 of Csikv\'ari (2026)},
which asserts the multiplicative Merino--Welsh inequality for simple and
cosimple matroids, and therefore also the stronger \emph{Conjecture~7.3}
therein, as observed by its author.  Our examples are rank truncations of direct sums of
$k$-element circuits.  We give a $3$-connected counterexample on $1011$
elements of rank $433$, certified by exact integer arithmetic, and a
$6048$-element counterexample admitting a proof without computation.
Taking the direct sum of a counterexample with its dual also gives a
negative answer to \emph{Problem~4.3 of Beke, Cs\'aji, Csikv\'ari, and
Pituk (2024)}: the original maximum and additive Merino--Welsh
inequalities can fail for simple and cosimple matroids.
The same construction disproves \emph{Conjecture~2 of Knauer,
Mart\'inez-Sandoval, and Merino (2026)}: for every fixed $g\ge3$,
$\Phi$ is unbounded even among matroids having both girth and cogirth
exactly $g$.  The dual direct-sum examples need not be connected;
these constructions do not disprove the graphical or binary versions.
\end{abstract}

\section{Introduction}

For a loopless and coloopless matroid $M$, write
\[
 \rho(M)=\frac{T_M(2,0)T_M(0,2)}{T_M(1,1)^2},
 \qquad
 \MW(M)=\frac1{\rho(M)}.
\]
The multiplicative Merino--Welsh inequality asserts $\rho(M)\ge1$.
Beke, Cs\'aji, Csikv\'ari, and Pituk disproved the unrestricted matroid
version using parallel thickenings of uniform matroids
\cite{BCCP}. Their published Problem~4.3 asks about the simultaneous
simple and cosimple restriction. Csikv\'ari explicitly states the
multiplicative version for this class as Conjecture~7.2 in
\cite{Csikvari}.

We resolve three named questions in the negative.  First, our highly
connected examples disprove \cite[Conjecture~7.2]{Csikvari} directly,
and thereby also disprove \cite[Conjecture~7.3]{Csikvari}, which its
author notes implies Conjecture~7.2.  The latter is an indirect
consequence: this manuscript does not construct an explicit bipartite
graph violating the permutation Tutte inequality.  Second, taking a direct sum with the dual also disproves the maximum
and additive inequalities for simple and cosimple matroids, answering
\cite[Problem~4.3]{BCCP}.  Third, the same dual direct sum has equal,
fixed girth and cogirth while its Merino--Welsh quotient is unbounded,
contradicting \cite[Conjecture~2]{KMM}.  These latter direct sums are
not asserted to remain connected.

Our construction replaces the two-element circuits underlying the
uniform-thickening examples by larger circuits, then truncates the
rank of their direct sum. Although a single $k$-element circuit has
$2^k-1$ independent sets, its $(2,0)$ Tutte evaluation is $2^k-2$.
The identity
\[
 (2^k-1)^2=2^k(2^k-2)+1
\]
allows the exponential number of bases in a suitable rank layer to
outgrow the product of the two axial evaluations.

\begin{theorem}\label{thm:main}
For every integer $k\ge3$, there is a sequence of $k$-connected
matroids $M_n$, representable over every infinite field, such that
\[
 g(M_n)=k,\qquad g^*(M_n)\longrightarrow\infty,\qquad
 \rho(M_n)\longrightarrow0.
\]
Consequently the multiplicative Merino--Welsh inequality fails even
for simple and cosimple, $3$-connected, rationally representable matroids.
\end{theorem}

All mathematical arguments needed for the counterexamples are given
below. The numerical search only supplies a smaller concrete parameter
choice; it is not used to prove the existence theorem. The literature
references identify the questions being answered and do not serve as
unproved inputs to the construction.

\section{Definitions and exact evaluations}

Let $C_k$ denote the circuit matroid on $k$ elements: its independent
sets are exactly the proper subsets of its ground set, and its rank is
$k-1$. This convention avoids any ambiguity in the ordering of the
two indices of a uniform matroid.

Fix pairwise disjoint sets $E_1,\ldots,E_n$, each of size $k$, and put
$E=\bigcup_i E_i$. Define
\[
 N_{k,n}=C_k^{\oplus n},\qquad s=(k-1)n.
\]
For $1\le r\le s$, let $M_{k,n,r}=\Tr_r(N_{k,n})$, where
$\Tr_r$ means truncation \emph{to} rank $r$. Explicitly,
\begin{equation}\label{eq:rank}
 \rk_M(A)=
 \min\left\{r,\sum_{i=1}^n\min\{|A\cap E_i|,k-1\}\right\}.
\end{equation}
Equivalently, the independent sets satisfy both
$|A|\le r$ and $|A\cap E_i|\le k-1$ for every $i$.
These are the usual independent sets of a truncated partition matroid;
in particular they define a matroid. One can also verify augmentation
directly: if $I,J$ obey the constraints and $|I|<|J|$, some block has
$|I\cap E_i|<|J\cap E_i|$, so an element of
$(J\setminus I)\cap E_i$ can be added to $I$.

We use the subset expansion
\begin{equation}\label{eq:tutte}
 T_M(x,y)=
 \sum_{A\subseteq E}
 (x-1)^{r-\rk_M(A)}(y-1)^{|A|-\rk_M(A)}.
\end{equation}
Introduce three polynomials for one block:
\begin{align}
 F_k(z)&=(1+z)^k-z^k, \label{eq:F}\\
 W_k(z)&=(1+z)^k-z^k-z^{k-1}, \label{eq:W}\\
 D_k(z)&=(1+z)^k-z^k+z^{k-1}. \label{eq:D}
\end{align}
Write $f_j=[z^j]F_k(z)^n$, $w_j=[z^j]W_k(z)^n$, and
$d_j=[z^j]D_k(z)^n$. Each of these coefficient arrays is nonnegative.

\begin{lemma}\label{lem:values}
For $M=M_{k,n,r}$,
\begin{align}
 T_M(1,1)&=f_r, \label{eq:bases}\\
 T_M(2,0)&=
 2\!\!\sum_{\substack{0\le j<r\\j\equiv r-1\;(\mathrm{mod}\;2)}}w_j,
 \label{eq:A}\\
 T_M(0,2)&=
 2^{kn}-
 2\!\!\sum_{\substack{0\le j<r\\j\equiv r-1\;(\mathrm{mod}\;2)}}d_j.
 \label{eq:B}
\end{align}
In particular,
\begin{equation}\label{eq:bounds}
 T_M(2,0)\le(2^k-2)^n,\qquad T_M(0,2)\le2^{kn}.
\end{equation}
\end{lemma}

\begin{proof}
The independent-set generating polynomial of one circuit is $F_k$.
A basis of $M$ is precisely an independent $r$-subset of $N_{k,n}$.
This proves \eqref{eq:bases}.

The signed rank generating polynomial of one block is $W_k$:
for ranks below $k-1$ its coefficients are $\binom{k}{j}$, whereas
at rank $k-1$ the $k$ subsets of size $k-1$ contribute positively and
the full block contributes negatively. Thus
\[
 W_k(z)^n=
 \sum_{A\subseteq E}(-1)^{|A|-\rk_N(A)}z^{\rk_N(A)}.
\]
Applying \eqref{eq:rank} in \eqref{eq:tutte} gives
\[
 T_M(2,0)=\sum_{j<r}w_j+
           \sum_{j\ge r}(-1)^{j-r}w_j.
\]
Since
\[
 W_k(1)=2^k-2,\qquad
 W_k(-1)=0,
\]
the second sum can be replaced by
$-\sum_{j<r}(-1)^{j-r}w_j$, proving \eqref{eq:A}.
The total sums of the even and odd coefficients of $W_k(z)^n$
both equal $(2^k-2)^n/2$. Nonnegativity now gives the first bound
in \eqref{eq:bounds}.

Similarly $d_j$ counts all subsets of $N_{k,n}$ having rank $j$:
the full block contributes one additional subset of rank $k-1$,
which explains $D_k$. Hence
\[
 T_M(0,2)=\sum_{j<r}(-1)^{r-j}d_j+\sum_{j\ge r}d_j.
\]
The total number of subsets is $\sum_jd_j=2^{kn}$; subtracting twice
the terms with negative sign proves \eqref{eq:B} and the second bound.
\end{proof}

\section{Unbounded quotients and connectivity}

For each $n$, choose $r_n$ to be an index of a largest coefficient of
$F_k(z)^n$, breaking ties arbitrarily. There are $s+1$ coefficients,
whose sum is $(2^k-1)^n$. Therefore
\begin{equation}\label{eq:pigeonhole}
 T_{M_{k,n,r_n}}(1,1)
 \ge \frac{(2^k-1)^n}{s+1}.
\end{equation}
Lemma~\ref{lem:values} implies
\begin{equation}\label{eq:rho}
 \rho(M_{k,n,r_n})
 \le (s+1)^2
 \left(\frac{2^k(2^k-2)}{(2^k-1)^2}\right)^n
 =(s+1)^2
 \left(1-\frac1{(2^k-1)^2}\right)^n.
\end{equation}
For fixed $k$ the last expression tends to zero.

We next check that this rank choice has all the required structural
properties; no assumption about an approximate location of a mode
is needed.

\begin{lemma}\label{lem:interior}
For fixed $k\ge3$, the maximizing indices above satisfy
$r_n\to\infty$ and $s-r_n\to\infty$.
\end{lemma}
\begin{proof}
Fix a nonnegative integer $L$. For $j\le L$,
\[
 [z^j]F_k(z)^n\le\binom{kn}{j}\le(kn)^L.
\]
There are $k^n$ bases of $N_{k,n}$. Every independent set of size
$s-d$, where $d\le L$, extends to one of these bases. Consequently
\[
 [z^{s-d}]F_k(z)^n
 \le k^n\binom{s}{d}\le k^n s^L.
\]
Both bounds are eventually smaller than $(2^k-1)^n/(s+1)$,
because $2^k-1>k$. A largest coefficient cannot occur at either
end within distance $L$. As $L$ was arbitrary, the conclusion follows.
\end{proof}

We use the standard connectivity function
$\lambda_M(A)=\rk_M(A)+\rk_M(E\setminus A)-r$.
A $j$-separation is a partition with both parts of size at least $j$
and $\lambda_M(A)\le j-1$. A matroid is $k$-connected if it has no
$j$-separation for $1\le j<k$.

\begin{lemma}\label{lem:structure}
If $r\ge k-1$ and $D:=s-r\ge k-1$, then $M_{k,n,r}$ is
$k$-connected, has girth $k$, and has cogirth
\begin{equation}\label{eq:cogirth}
 g^*(M)=kn-(r-1)-
        \left\lfloor\frac{r-1}{k-1}\right\rfloor
 =D+1+\left\lceil\frac{D+1}{k-1}\right\rceil.
\end{equation}
\end{lemma}
\begin{proof}
Every set of fewer than $k$ elements is independent in $N_{k,n}$
and remains independent after truncation. Each full block $E_i$
remains a circuit. Thus $g(M)=k$.

Fix $1\le j<k$ and a partition $E=A\sqcup B$ with
$|A|,|B|\ge j$. Put $a=\rk_N(A)$ and $b=\rk_N(B)$.
Since the girth of $N$ is $k$, we have $a,b\ge j$.
Also $a+b\ge s$. If $a,b<r$, then
\[
 \lambda_M(A)=a+b-r\ge s-r=D\ge k-1\ge j.
\]
If, say, $a\ge r$, then
$\lambda_M(A)=\min(b,r)\ge j$.
This excludes every such separation, proving $k$-connectivity.

For the cogirth formula, consider a subset $H$ with
$\rk_M(H)\le r-1$. Its rank in $N$ is then at most $r-1$.
A block contributes at most its rank plus one to $|H|$, and the
extra element is available only if the block is full, which costs
$k-1$ units of rank. Therefore
\[
 |H|\le(r-1)+\left\lfloor\frac{r-1}{k-1}\right\rfloor.
\]
This bound is attained by taking as many full blocks as possible
at rank budget $r-1$, followed by the required proper subset of one
further block. The resulting set is a hyperplane of the truncation.
The complements of hyperplanes are exactly the cocircuits, giving
the first formula in \eqref{eq:cogirth}. The second follows from
$s=(k-1)n$ and $D=s-r$.
\end{proof}

\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
For sufficiently large $n$, Lemma~\ref{lem:interior} ensures the
hypotheses of Lemma~\ref{lem:structure}. Formula~\eqref{eq:cogirth}
also shows $g^*(M_n)\to\infty$, and \eqref{eq:rho} shows
$\rho(M_n)\to0$.

Let $\mathbb F$ be any infinite field. Each $C_k$ has a representation
using a basis $e_1,\ldots,e_{k-1}$ together with
$e_1+\cdots+e_{k-1}$. The direct sum gives a representation of $N$.
A sufficiently generic linear map $\mathbb F^s\to\mathbb F^r$
preserves independence of every originally independent set of size
at most $r$. Such a map exists: for each of finitely many independent
sets, some minor of its projected columns is a nonzero polynomial
in the entries of the map; the product of these polynomials is nonzero
and cannot vanish everywhere over an infinite field.
For sets of larger rank, an independent $r$-subset survives, while
no rank can exceed $r$. Thus the projected rank is exactly
\eqref{eq:rank}. This proves representability over $\mathbb F$.
\end{proof}

\begin{remark}
These examples are also laminar matroids: their independent sets are
specified by capacities on the disjoint blocks $E_i$ and on their
union $E$, a laminar family. This assertion follows directly from
the displayed independent-set description.
\end{remark}

\section{A concrete counterexample and exact verification}

Take $k=3$, $n=337$, and $r=433$:
\[
 M=\Tr_{433}(C_3^{\oplus337}).
\]
The ground set has $1011$ elements, the rank is $433$, and the corank
is $578$. Since $s-r=674-433=241\ge2$, the matroid is
$3$-connected. Its girth is $3$ and \eqref{eq:cogirth} gives
$g^*(M)=363$. In particular, it is simple and cosimple.

For $k=3$, the three one-block polynomials are
\[
 F_3(z)=1+3z+3z^2,\quad
 W_3(z)=1+3z+2z^2,\quad
 D_3(z)=1+3z+4z^2.
\]
Applying Lemma~\ref{lem:values} with exact integer arithmetic gives
\[
 \rho(M)=0.987478184929103952874888526897\ldots<1.
\]
A shorter certificate uses only
$b=[z^{433}](1+3z+3z^2)^{337}$:
\begin{equation}\label{eq:finite-cert}
 b^2>48^{337}\ge T_M(2,0)T_M(0,2).
\end{equation}
The following self-contained Python code verifies the strict integer
comparison in \eqref{eq:finite-cert}.
\begin{verbatim}
c = [1]
for _ in range(337):
    d = [0] * (len(c) + 2)
    for j, value in enumerate(c):
        d[j] += value
        d[j + 1] += 3 * value
        d[j + 2] += 3 * value
    c = d
assert c[433] ** 2 > 48 ** 337
\end{verbatim}

A separate verification script \texttt{verify\_simple\_cosimple\_mw.py}
computes all three Tutte evaluations and writes their full integer
values to \texttt{certificate\_n337\_r433.json}.  The source archive
for this manuscript is self-contained and does not require the script.
All inequality decisions use integers. Decimal ratios are displayed
only for readability.

Two verification routes were used. First, independent coefficient
implementations agreed on the $1011$-element example. Second, for
$1\le n\le4$ and every $1\le r\le2n$ (20 parameter pairs),
direct enumeration of every subset using \eqref{eq:rank} agreed with
all three coefficient formulas. An optional exhaustive search within
this family first finds a failure at $n=337$, at ranks $433$, $434$,
and $435$, after checking $113232$ parameter pairs. This is a
family-specific search result, not a claim of minimality among all
simple and cosimple matroids.

\section{Consequences and scope}

\subsection*{The permutation Tutte formulation}
Csikv\'ari \cite[Conjecture~7.3]{Csikvari} conjectures a multiplicative
permutation Tutte inequality for bipartite graphs of minimum degree at
least two, and observes that it implies his Conjecture~7.2.  Our
counterexample to Conjecture~7.2 therefore also implies that
Conjecture~7.3 is false.  This is a logical consequence, rather than
an explicit bipartite-graph construction.

\subsection*{The maximum and additive versions}
Let $M$ be any one of the simple and cosimple multiplicative
counterexamples, and put $Q=M\oplus M^*$. Direct sum and duality give
\[
 T_Q(2,0)=T_Q(0,2)=T_M(2,0)T_M(0,2)
 <T_M(1,1)^2=T_Q(1,1).
\]
Thus $Q$ is simple and cosimple and violates both the maximum and
additive versions as well. This answers the simple and cosimple
question in \cite[Problem~4.3]{BCCP}, alongside the explicit
multiplicative formulation in \cite[Conjecture~7.2]{Csikvari}.
The direct sum $Q$ is disconnected; we do not claim these latter
two versions fail under arbitrarily high connectivity.

\subsection*{Fixed girth and cogirth}
Knauer, Mart\'inez-Sandoval, and Merino
\cite[Conjecture~2]{KMM} ask for a finite upper bound on $\MW(M)$
depending only on a common girth and cogirth parameter $g\ge3$.
Fix $k\ge3$ and take $M_n$ from Theorem~\ref{thm:main}. Set
$Q_n=M_n\oplus M_n^*$. For all sufficiently large $n$,
\[
 g(Q_n)=g^*(Q_n)=k,\qquad
 \MW(Q_n)=\MW(M_n)^2\longrightarrow\infty.
\]
This gives a negative answer even when both girth and cogirth
are required to be \emph{exactly} the prescribed value.

\subsection*{Relation to graphical and binary matroids}
The original direct sum $N_{k,n}$ is graphic, but its truncation
need not be. In the $1011$-element example, keep one element from
each of four blocks. In the other $333$ blocks choose an independent
set $C$ of size $431$, contract $C$, and delete every element except
the four kept elements. The resulting rank function is
\[
 \rk(X)=\min(2,|X|),
\]
so the minor is the four-element rank-two uniform matroid $U_{2,4}$.
It is not binary: a two-dimensional binary vector space has only
three nonzero vectors. Thus this example is neither binary nor
graphic, and it does not settle those separate restrictions.

\section*{Appendix: a finite certificate without computation}

There is also a completely explicit parameter choice whose failure
can be checked by hand:
\[
 k=3,\qquad n=2016,\qquad r=2592.
\]
This gives $6048$ elements. Consider only independent sets with
exactly $288$ empty blocks, $864$ blocks of occupancy one, and
$864$ blocks of occupancy two. Their total size is $2592$, and
their number is
\[
 L=\frac{2016!}{288!\,864!\,864!}\,3^{1728}
 \le [z^{2592}](1+3z+3z^2)^{2016}.
\]
We claim
\begin{equation}\label{eq:type-lower}
 L\ge\frac{7^{2016}}{\binom{2018}{2}}.
\end{equation}
Use the multinomial distribution with $2016$ trials and probabilities
$(1/7,3/7,3/7)$. The count vector $(288,864,864)$ is its integer
mean and is a mode. To verify the latter assertion directly, write
this vector as $(a_i)_{i=0}^2$ and let $(m_i)_{i=0}^2$ be any
other count vector with the same sum. The ratio of the probability
at $(m_i)$ to the probability at $(a_i)$ is
\[
 \prod_{i=0}^2 \frac{a_i!\,a_i^{m_i-a_i}}{m_i!}\le1.
\]
Each factor is at most one: for an increase it is a product of
$a_i/(a_i+j)$, and for a decrease it is a product of
$(a_i-j)/a_i$. There are $\binom{2018}{2}$ possible count vectors,
so the mode has probability at least its reciprocal. Its probability
is exactly $L/7^{2016}$, proving \eqref{eq:type-lower}.

Finally, the binomial theorem gives
\[
 \left(\frac{49}{48}\right)^{48}
 =\left(1+\frac1{48}\right)^{48}>2,
 \qquad
 \binom{2018}{2}=2035153<2097152=2^{21}.
\]
Since $2016=48\cdot42$, it follows that
\[
 \left(\frac{49}{48}\right)^{2016}
 >2^{42}>\binom{2018}{2}^{\,2}.
\]
Writing $b=[z^{2592}](1+3z+3z^2)^{2016}$, we conclude
\[
 b^2\ge\frac{49^{2016}}{\binom{2018}{2}^{\,2}}
 >48^{2016}\ge T_M(2,0)T_M(0,2).
\]
Its rank is at least two and its truncation deficit is
$4032-2592=1440$, so it is $3$-connected, simple, and cosimple
by Lemma~\ref{lem:structure}. This finite proof requires neither
a numerical search nor a floating-point estimate.

\begin{thebibliography}{9}
\bibitem{BCCP}
C. Beke, G. K. Cs\'aji, P. Csikv\'ari, and S. Pituk,
\emph{The Merino--Welsh conjecture is false for matroids},
Advances in Mathematics \textbf{446} (2024), 109674.
\href{https://doi.org/10.1016/j.aim.2024.109674}
{doi:10.1016/j.aim.2024.109674}.
The published version contains Problem~4.3.

\bibitem{Csikvari}
P. Csikv\'ari,
\emph{Around the Merino--Welsh conjecture: improving Jackson's inequality},
European Journal of Combinatorics \textbf{137} (2026), 104402.
\href{https://doi.org/10.1016/j.ejc.2026.104402}{doi:10.1016/j.ejc.2026.104402}.
Conjecture~7.2 is stated in
\href{https://arxiv.org/abs/2502.19196}{arXiv:2502.19196v2}.

\bibitem{KMM}
K. Knauer, L. Mart\'inez-Sandoval, and C. Merino,
\emph{Merino--Welsh inequalities for matroids with controlled lattices of
cyclic flats},
\href{https://arxiv.org/abs/2609.38047}{arXiv:2609.38047},
version 1, 29 September 2026. Conjecture~2.
\end{thebibliography}
\end{document}