36 problems

Hamiltonian path pattern conjecture. In order for a Hamiltonian path to exist on $I[G]$, it is necessary and sufficient that $I[G]$ satisfy one of the four possible patterns:

Subtree separation conjecture. The universal support-forest-counting combinations arising from $M_r^{(2)}(T)$, together with the Laplacian spectrum, determine all embedded forest c…

Spectral support-forest-profile conjecture. If $T_1$ and $T_2$ have the same Laplacian spectrum and the same $(n-2,2)$ support-forest profile, then $T_1$ and $T_2$ are isomorphic.…

Lifting conjecture. Then, for some $S$,

Let $C_{n_1}(R_1)$ and $C_{n_2}(R_2)$ be circulant graphs such that … for some $R$. Product criterion conjecture. The graph $C_{n_1n_2}(R)$ has Type-2 isomorphic circulant graphs i…

Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be two distinct immersion-closed and union-closed graph classes. For graphs $G$ and $H$, write $G =_{\mathcal{M}} H$ when…

Let $\mathcal{M}$ be a graph class that is proper, minor-closed, and union-closed. For graphs $G$ and $H$, write $G =_{\mathcal{M}} H$ when $\hom(K,G)=\hom(K,H)$ for every…

Let $G$ and $H$ be trees, and let $\chi_G(x)$ denote the chromatic symmetric function of $G$. Tree isomorphism conjecture. If $G$ and $H$ are non-isomorphic trees, then … The conje…

For a graph $G$, its Hadwiger number is the largest integer $t$ such that $G$ contains the complete graph $K_t$ as a minor. Two graphs are homomorphism indistinguishable over a gra…

Let $G$ be a graph, and let $fbar{X}_G$ denote its Kromatic symmetric function. Two graphs are isomorphic when they differ only by a relabeling of their vertices. Pierson's conject…

Let $G,H,G_a,G_b$ be graphs, let $E_G\subset E(G)$ and $E_H\subset E(H)$ be edge subsets, let $T_a,T_b$ be spanning trees, and let $S_G\in V(G)\cup E(G)$ and $S_H\in V(H)\cup E(H)$…

Let $D(q;m,n)=D_2(\mathbb F_q;X^mY^n)$ be a monomial digraph over the field with $q$ elements. Kodess's conjecture. For a prime power $q$, the digraphs $D(q;m_1,n_1)$ and…

Let $G$ be a graph on $n$ vertices, with distinct degrees $\delta_1 > \delta_2 > \cdots > \delta_p$. Set $r = \lceil\log_2 n\rceil + 1$, partition $V = \bigsqcup_{i=1}^r V_i$ by ta…

Let $\Sigma_1$ and $\Sigma_2$ be signed complete graphs. Write $\Sigma_1\cong\Sigma_2$ when they are isomorphic, and let $\mathscr B(\Sigma,x,y)$ and $\mathsf E(\Sigma,x,y)$ denote…

Asymptotic count conjecture. As $v \to \infty$, the counts $\alpha(v)$ and $\beta(v)$ are respectively asymptotic to

Let $C_{11}$ denote the cycle on eleven vertices. A Šoltés graph is a graph for which deleting any vertex leaves the Wiener index unchanged. Šoltés graph uniqueness conjecture. The…

GI-hardness conjecture. $\textsc{Comp-Sub}(\mathcal{PM})$ is $\GI$-hard on $C_{k\geq 6}$-free graphs.

Finite principal-specialization conjecture. For all such trees,

Stanley's chromatic conjecture for trees. Trees $T$ and $U$ are isomorphic as graphs if and only if

High-ordered spectral characterization conjecture. Two trees are isomorphic if and only if they have the same high-ordered spectrum.

Let $l,m,n,r otinmathbb{N}$?

Triangle-free graph conjecture. If

Let $G$ be a graph, let $(y,x)\in P^G_A$ be a vector, and let … be a fixed shrinking strategy. Shrinking-strategy isomorphism conjecture. The graphs obtained by applying…

Lovász's planar homomorphism question. If

Spectral determination conjecture. Almost every graph is determined by its spectrum.