58 problems

El-Zahar's conjecture. If

Let $a$ and $b$ be positive integers with $a\le b$, and let $G$ be a finite undirected simple connected graph of order $n$ satisfying … For a graph $H$, write $\rho(H)$ for its spe…

Let $r\ge 2$ and $k\ge 0$, and let $G$ be a connected graph of order $n\ge 2(2r+k+2)(r+k+2)$ with minimum degree $\delta(G)\ge r+k$. Let $F_n^{r,k}$ be the extremal graph appearing…

Let $F$ be a strictly $1$-balanced graph, and let $p^*$ denote the relevant sharp threshold for the disappearance of $F$-isolated vertices. Johansson–Kahn–Vu threshold conjecture.…

Let $r\geq 2$, and let $G$ be an $((r-1)n+1)$-regular graph on $rn$ vertices. A subset of $V(G)$ induces a $K_r$-factor when the graph induced by that subset contains a vertex-disj…

Let $\boldsymbol{G}_{d_1+d_2}\sim\mathcal{G}(n,d_1+d_2)$, and let $\mathcal{R}_{d_1}(G)$ be the set of $d_1$-regular spanning subgraphs of $G$. For $d_1,d_2>0$ with…

Let $k\geq 3$ be an integer, let $G$ be a graph, and for a vertex set $X\subseteq V(G)$ let $c_{2j+1}(G-X)$ denote the number of components of $G-X$ having order $2j+1$. A {…

For a fixed integer $k \geq 4$, let a $TT_k$-factor be a collection of vertex-disjoint copies of the transitive tournament $TT_k$ covering all vertices, and let an almost $TT_k$-fa…

Let $G$ be an $n$-vertex graph, and let $b\ge a$ be two positive integers such that $na\equiv 0\pmod{2}$ and $n\ge a+1$. Here, $\lambda(G)$ denotes the spectral radius of $G$,…

Let $G=(V,E,w)$ be an edge-weighted complete graph on $n$ vertices, where $w:E\to[0,1]$. For $v\in V$, define its weighted degree by $d^w(v):=\sum_{uv\in E}w(uv)$ and let…

Let $k$ be a positive integer and let $G$ be a graph of order $n$. For an independent set of order $k+1$, define … when $\alpha(G)\geq k+1$, and set $\sigma_{k+1}(G)=\infty$ otherw…

Let $r\geq 2$, let $n$ be divisible by $r$, and let $G$ be an $(\tfrac{(r-1)n}{r}+1)$-regular graph on $n$ vertices. A subset of $V(G)$ induces a $K_r$-factor if its induced subgra…

Let $F$ be a strictly $1$-balanced graph, and consider the threshold for the existence of an $F$-factor and the threshold for the disappearance of $F$-isolated vertices. Ruciński's…

Rao–Rao conjecture. A graphic degree sequence $(d_1,d_2,\ldots,d_n)$ is $k$-factorable if and only if the sequence $(d_1-k,d_2-k,\ldots,d_n-k)$ is graphic.

Let $G(n,p)$ be the binomial random graph, and let $K_r$ be the complete graph on $r$ vertices. A vertex is $K_r$-isolated if it is not contained in any copy of $K_r$, and let…

Let $G$ be a cyclically $5$-edge-connected odd $2$-factored snark, where a snark is a bridgeless cubic graph of chromatic index four and odd 2-factored means that every cycle in ev…

Let $G$ be a 3-edge-connected pseudo $2$-factor isomorphic cubic bipartite graph, and suppose that $G=G_1*G_2$ is a star-product decomposition. Abreu–Diwan–Jackson–Labbate–Sheehan'…

Let $G$ be an essentially $4$-edge-connected pseudo $2$-factor isomorphic cubic bipartite graph, where essentially $4$-edge-connected means 3-edge-connected with no non-trivial 3-e…

Let $G$ be a 3-edge-connected cubic bipartite graph. A graph is pseudo 2-factor isomorphic when the parity of the number of circuits is the same for all its $2$-factors. Let…

Let $K_5$ denote the complete graph on five vertices, and let a graph be 2-factor Hamiltonian when every $2$-factor is a Hamiltonian circuit. Abreu–Aldred–Funk–Jackson–Sheehan's co…

Let $G$ be a 3-edge-connected 2-factor isomorphic cubic bipartite graph. Abreu–Diwan–Jackson–Labbate–Sheehan's conjecture. Then $G$ is a 2-factor Hamiltonian cubic bipartite graph.…

Let $G$ be a connected $k$-regular bipartite graph. A graph is 2-factor isomorphic when all its $2$-factors are isomorphic, and it is 2-factor Hamiltonian when every $2$-factor is…

Let $G$ be a $2$-factor Hamiltonian $k$-regular bipartite graph, meaning that every $2$-factor of $G$ is a Hamiltonian circuit. Let $K_{3,3}$ and $H_0$ be the two specified cubic b…

Let $k\geq 2$ be an integer, and let $n$ be a sufficiently large multiple of $k$. Let … be a collection of graphs on a common vertex set of size $n$. Write $\delta(\bm{G})$ for the…

Let $H$ be a graph, let $d^*$ denote its maximal density parameter, and let $F_H(alpha n,n)$ be the minimal weight of an $(alpha,H)$-factor in the random weighted complete graph…