21 problems

Whiteley's conjecture. These matroids are equal for every $n\geq 2$:

The three-dimensional generic rigidity matroid is the matroid associated with generic rigidity in three-dimensional space, and Whiteley's cofactor matroid is the unique maximal mat…

Whiteley's maximality conjecture. For all $n,d$, ${\mathcal C}^{d-2}_{d-1}(K_n)$ is the unique maximal abstract $d$-rigidity matroid on $K_n$.

The Dress Conjecture. The rank of $G$ satisfies

Lovász–Yemini conjecture. There exists an integer $c_d$, possibly $c_d=d(d+1)$, such that every $c_d$-connected graph is $d$-rigid.

Jackson–Jordán's conjecture. Every 5-regular, 3-sparse graph is $\mathcal{R}_3$-independent. This is the remaining case of the Jackson–Jordán characterization for graphs with bound…

Let $D$ be a double ${\mathcal R}_3$-circuit, and let $w$ be a technicolour vertex of degree $5$ in $D$. The principal partition of $D$ is the partition associated with its double-…

A $3$-dimensional $X$-replacement replaces two non-adjacent edges $uv$ and $xy$ by a new vertex $w$ adjacent to $u,v,x,y$ and having degree $5$. A graph is minimally…

For integers $k\geq 2$ and sufficiently large $n$, consider the rigidity matroid ${\mathcal R}_{2k+1}(K_n)$ of the complete graph $K_n$ in dimension $2k+1$. The matroid matching co…

Clinch–Jackson–Tanigawa's maximality conjecture. For all $n,d$, ${\mathcal C}^{d-2}_{d-1}(K_n)$ is the unique maximal $K_{d+2}$-matroid on $K_n$.

Let $G$ be a bipartite graph, let $d\geq 0$ be an integer, and let $\mathcal{B}_{m,n}(d,d)$ denote the matroid introduced in the paper. A $d$-coloring of the edges of $G$ is $d$-Be…

Let $d$ be a positive integer, let $G$ be an $\mathcal{R}_d$-connected graph, and let $e$ be an edge of $G$. An edge is an $\mathcal{R}_{d+1}$-bridge when its deletion lowers the r…

Let $G$ be a graph and let $d$ be a positive integer. A graph is redundantly $\mathcal{R}_d$-connected when deleting any edge leaves it $\mathcal{R}_d$-connected. The connectivity…

Kalai–Whiteley maximality conjecture. The matroid $\mathcal{H}_d(n)$ is the freest matroid in which every $K_{d+2}$ and every $K_{d+1,d+1}$ are circuits.

Kalai–Whiteley conjecture. For every $n$ and $d$,

Let $G=(V,E)$, $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be graphs. Say that $G$ is a $t$-sum of $G_1,G_2$ along an edge $e$ when … Here ${\mathcal R}_d$ denotes the $d$-dimensional rigi…

Let $K_n$ be the complete graph and let $F\subseteq E(K_n)$. A proper $\{K_6,K_{6,6}\}$-sequence is the sequence construction defined in the source using copies of $K_6$ and…

Let $M_0$ be the rank ${{d+2}\choose{2}}$ truncation of ${\cal R}_{d,n}$, equivalently of any $K_{d+2}$-matroid on $E(K_n)$, with $n\geq d+2$. A $K_{d+2}$-sequence is a sequence of…

Let $F\subseteq E(K_n)$ be a flat in ${\cal R}_{3,n}$. Let ${\cal X}^*$ be the family of maximal cliques in the graph $K_n[F]$ having at least three vertices, and let $F_0$ consist…

Let $K_n$ be the complete graph, let $F\subseteq E(K_n)$, and let ${\cal X}$ be a 2-thin cover. Define ${\rm val}_{\rm D}({\cal X})$ by … A family is 9-degenerate if its members ca…

Let $H$ be a graph, and let $e_1,e_2$ and $e_1',e_2'$ be two pairs of adjacent edges such that $H\cup\{e_1,e_2\}$ and $H\cup\{e_1',e_2'\}$ are both $C_2^1$-independent, with the co…