43 problems

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…

El-Zahar's conjecture. If

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 $G$ 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 $r\geq 2$, and let $G$ be an $((r-1)n+1)$-regular graph on $rn$ vertices. A subset $S\subseteq V(G)$ is said to induce a $K_r$-factor when $G[S]$ contains a collection of verte…

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 $G$ 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 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 $K_5$ 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. Sheehan's conjecture. There are no $2$-factor Hamilt…

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 $G$ be a finite simple $k$-regular graph, where a graph is uniquely Hamiltonian if it has exactly one Hamiltonian circuit. Sheehan's conjecture. There are no uniquely Hamiltoni…

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 $\boldsymbol{G}_{d_1+d_2}\sim\mathcal{G}(n,d_1+d_2)$, 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 $H$ 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…

The asymptotic $H$-factor conjecture. Given $\mu>0$, $h\in\mathbb{N}$, and an $h$-vertex graph $H$ with $\mathrm{hcf}(H,\mathrm{AP})=1$, there exists $\alpha>0$ such that, for all…

Non-central-factor conjecture. Let $H$ be a fixed non-central graph. Then, for every strategy $\sigma\in\operatorname{\mathcal{S}}_n$,

Average minimum-degree conjecture. For every $k\geq3$ and $\varepsilon>0$, there exists $n_0=n_0(k,\varepsilon)$ such that for every $n\geq n_0$, if there are numbers…

Häggkvist's conjecture. For every $k\geq 3$, if

Let $k$ and $n$ be positive integers such that … are both even integers. Let $T$ be a tree of order $k$, and let $c:E(K_n)\to\{-1,1\}$ be a zero-sum labeling of the complete graph…

Let $b_\Delta$ denote the threshold bias for the biased Maker–Breaker $H$-factor game, where $H$ has maximum degree $\Delta$. Threshold conjecture. For all $\Delta \ge 3$, … This p…