11 problems

Let $G$ be a $d$-regular bipartite graph on $v(G)$ vertices, and let $m_k(G)$ denote the number of matchings of size $k$. Set $p=2k/v(G)$, and let $\mathbb{G}_d(p)$ denote the stat…

The $D_n$ conjecture. There exists an integer $n_0\geq 8$ such that, if $n\geq n_0$, then $\widetilde H_{\nu_{2n}}(M(D_n))$ is an elementary $3$-group. Moreover, if $n\geq n_0$ and…

Let $k\geq 3$, and let $G$ be a graph that is $\{C_4,C_6,C_{2k}\}$-free. Lin–Zeng's conjecture. The existence of a perfect matching in the hypotheses of their bisection result can…

Structural matching conjecture. With high probability as $n\rightarrow\infty$, the graph $G-S_r(G)$ contains a $K_{r-1}$-matching that covers all but $o(\sqrt{\mu_r})$ vertices of…

Let $G$ and $H$ be graphs with planar union, let $\mathcal{PM}(G,H)$ be the matchings pairing edges from $G$ and $H$, and let $s_M$ be the leading coefficient obtained by summing…

Let $G$ be a $3$-edge-connected graph. A matching is a set of pairwise vertex-disjoint edges, and a 3-edge-cut is an edge cut of size three. A matching is deletable when all its ed…

Hereditary characterization conjecture. For every positive integer $k$, the set ${\cal G}_k$ equals the set of all graphs $G$ such that

Let $P_3$ be the path on three vertices, let $k$ be a positive integer, and let $ex(n,H)$ denote the maximum number of edges in an $n$-vertex graph containing no subgraph isomorphi…

For the Federbush expansion of the matching entropy, write the coefficients as $a_k$. For the square lattice $\mathbb{Z}^2$, the computed values include … Equivalently, for the mod…

Let $L$ be a random rooted graph with spectral measure $\rho_L$ absolutely continuous with respect to Lebesgue measure, with density $f(z)$ satisfying … for some $varepsilon$ and…

Let $G$ be the random $(a,b)$-biregular bipartite graph considered in the paper, and let $m_k(G)$ denote the number of matchings of size $k$. Second-moment conjecture. There exists…