15 problems

Let $X_1,X_2$ and $X_3$ be the partite sets of the complete tripartite graph $K_{n_1,n_2,n_3}$, where $|X_i|=n_i$ for $i\in \llbracket 3\rrbracket$. Let $F$ be a spanning forest in…

Lin et al.'s conjecture. The complete $k$-partite graph $K_{n_{1},n_{2},\ldots,n_{k}}$ is determined by its $D$-spectrum.

Let $n \geq 3$, let $1 \leq a_1 \leq \cdots \leq a_n$, and define … If $a_1 + \cdots + a_{n-1} > 6$ and $a_n \geq e$, then the family-size stabilization conjecture asserts that … H…

Let $r$ be the number of parts, and let $K_{2,\ldots,2}$ be the complete multipartite graph with $r$ parts, each of size $2$. For a graph $H$, write $sat(n,H)$ for the minimum numb…

Let $G=K_{n_1,n_2,\ldots,n_t}$ be a complete $t$-partite graph such that … and $n_i\geq 2$ for all $i$. A $\mathbb{Z}_{2^n}$-distance-magic labeling is a bijection from the vertice…

Mahmoodian–Mirzakhani conjecture. These necessary conditions are also sufficient for $K_{r,s,t}$ to decompose into 5-cycles. This conjecture gives necessary and sufficient conditio…

For positive integers $m$, $a$, and $k$, let $K^{(m)}_a$ be the complete $m$-partite graph with $a$ vertices in each partite set. Let $D(G,k)$ denote the minimum defect in a fracti…

Let $M(K_{1(n),m_{\ell},m_{\ell-1},\ldots,m_2,m_1})$ denote the IC-index considered in Proposition 3.3, with $\ell$, $m_{\ell}$, and $n$ as in that proposition. Equality conjecture…

Crossing-number equality conjecture.

Rectilinear crossing-number conjecture.

Let $K_{a_1,a_2,a_3,n}$ denote the complete multipartite graph whose parts have sizes $a_1,a_2,a_3,n$, let $K_{2,4,n}$ denote the complete multipartite graph with parts of sizes…

Let $K_{k,m,n}$ be the complete tripartite graph with part sizes $k,m,n$, where $k\leq m\leq n$ and $n>k+m$. Reduction conjecture. … This is proposed as a generalization of the…

Let $K_{1,m,n}$ be the complete tripartite graph with part sizes $1,m,n$, and let an interval $t$-coloring mean an interval edge-coloring using the colors $1,\ldots,t$. Exact-spect…

Let $G$ be a complete multipartite graph, let $k$ be a positive integer, and let $S$ be a $k$-subset of $V(G)$. Li's conjecture. We have $\kappa_k(G)=\kappa(S)$ when the numbers of…

Li's conjecture. We have $\kappa_k(G)=\kappa(S)$. This is one of two conjectures proposed by W. Li in her Ph.D. thesis; the supplied text gives no resolution status.