160 problems

Let $G$ be a connected graph on $n\geq 3$ vertices, let $k$ be its matching number, and let $\rho(G)$ denote its spectral radius. Aouchiche–Hansen conjecture. Based on extensive co…

Aouchiche–Hansen–Zheng conjecture. One has

Let $[n]=\{1,2,\ldots,n\}$, let $\binom{[n]}{k}$ denote the family of all $k$-subsets of $[n]$, and let $\mathcal{F}\subseteq\binom{[n]}{k}$ be a $k$-uniform hypergraph. Its matchi…

Randić-index matching conjecture. If

Lou–Yu conjecture. The minimum degree of $G$ is one of

A rainbow matching is a matching whose edges have pairwise distinct colors. Grinblat's conjecture. If $G$ is a multigraph that is not necessarily properly edge colored with $n$ col…

Erdős matching conjecture. For all $n\geq (s+1)k-1$, if $\nu(\mathcal{H})\leq s$, then

Erdős matching conjecture. The number of edges of $H$ satisfies

Let $G$ be a finite simple graph with $n$ non-isolated vertices. Let $L(G)$ be its Laplacian matrix, let $\lambda_1(G)\ge\cdots\ge\lambda_n(G)$ be the Laplacian eigenvalues, and wr…

Carvalho–Lucchesi–Murty conjecture. There exists a non-trivial ELP-cut $D$ that is laminar with $C$.

For integers $N,k,t$ with $Nge kt$, let $e_k(N,t)$ be the maximum size of a $k$-uniform family in $\binom{[N]}k$ with matching number less than $t$. Erdős's matching conjecture. ……

Erdős–Kleitman conjecture. If $\mathcal{F}$ contains no matching of size $s$ and is maximal with respect to this property, then

Let $G$ be a finite simple graph. A connected matching is a matching $M\subseteq E(G)$ such that for every two edges $e,f\in M$, an endpoint of $e$ is adjacent to an endpoint of…

Norine–Thomas's conjecture. Every minimal brick $G$ has at least

Balister–Győri–Schelp conjecture. There exists a partition of $\mathbb F_2^k$ into 2-sets $\{w_i,z_i\}$, for $i\in\{1,\dots,2^{k-1}\}$, such that

3-extendability conjecture. Every $8$-regular optimal $1$-embedded graph $G$ on the Klein bottle is $3$-extendable.

Let $\mathcal{L}\in\mathcal{D}^k$ be a family of $k$ matroids on $V$, and let $w\in\mathbb{R}_{\ge0}^V$. Weighted fractional matching-cover conjecture. One has … This is presented…

Erdős's matching conjecture. If $\mathcal{F}\subset\binom{[n]}{k}$ satisfies $\nu(\mathcal{F})\leqslant s$, then

Larson–Pepper's conjecture. The equality $\alpha=a$ holds if and only if $G$ is a König–Egerváry graph and every maximum independent set of $G$ is a maximal annihilating set.

Assume there are $n$ representatives of each of three genders, with complete cyclic preference lists: representatives of the first gender rank representatives of the second, repres…

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each part, and partition its edge set into $n$ classes $E_1,\ldots,E_n$, each of size $n$. Brualdi–Stein colorful…

Let $H$ be a $k$-uniform hypergraph on $n$ vertices, let $e(H)$ denote its number of edges, and let $u(H)$ denote its matching number, the maximum number of pairwise disjoint edge…

Let $G$ be a finite simple graph. It is factor-critical if, for every vertex $v\in V(G)$, the graph $G-v$ has a perfect matching. The graph $G$ is $K_{1,5}$-free if it has no induc…

The Asymptotic Lower $r$-Permanent Conjecture. Under these hypotheses,

Let $\{G_k=(V_k,E_k)\}_{k\in\mathbb{N}}$ be a sequence of $r$-regular bipartite graphs with $\#V_k\to\infty$. Let $h_{\{G_k\}}(p)$ denote the associated monomer–dimer entropy, and…