102 problems

Stanley--Gasharov conjecture (1998). Every claw-free graph is Schur-positive.

Let $c_q(G)$ denote the number of proper $q$-colorings of a graph $G$. Let $G$ be an $n$-vertex $d$-regular graph, with $q\ge 3$, and let $K_{d,d}$ be the complete bipartite graph…

Second-range formula conjecture. Then

Let $K_p$ be the complete graph on $p$ vertices, and let $f(n,H)$ be the smallest number of colors in an edge coloring of $K_n$ in which every copy of $H$ intersects at least one c…

Exact-extremal-set conjecture. Let $n,r$ be positive integers with $r\geq4$. If $n$ is even and $r\leq7$, then

Geodesic Norine conjecture. For $n\ge 2$, every antipodal edge-coloring of $Q_n$ contains antipodal vertices $u$ and $\bar u$ joined by a monochromatic geodesic path.

Let $H$ be a graph that is $k$-colorable. For a finite graph $G$, let ${\rm Hom}(G,H)$ denote the space of graph homomorphisms from $G$ to $H$, with adjacency given by changing one…

Bradač–Liu–Wu–Xu conjecture. For every integer $k\geq 1$,

Let $M(H,n)$ be the minimum number of monochromatic copies of a graph $H$ in a red/blue coloring of $K_n$, and define the threshold Ramsey multiplicity by … where $r(H)$ is the Ram…

Let $k$ be a positive integer, let $c_k=c(k)$ be an integer, and let $G$ be a graph on $n$ vertices such that $k\mid E(G)$. The composite-modulus linear-bound conjecture. There is…

Let $p$ be prime, let $c_p=c(p)$ be an integer, and let $G$ be a graph on $n$ vertices such that $p\mid E(G)$. The graph linear-bound conjecture. There is an integer $c_p=c(p)$ suc…

Let $p$ be prime, let $n_p=n(p)$ be a positive integer, and let $F$ be a forest on $n\geq n_p$ vertices such that $p\mid e(F)$. Here $R(F,\mathbb{Z}_p)$ denotes the least order of…

Let $\mathbf{G}$ be the set of isomorphism classes of finite simple graphs. For $G\in\mathbf{G}$, let $\varphi(G)$ be the graph invariant defined by … where $\chi_3(G)$ is the numb…

Let $G$ be a graph, and let $\widetilde{\chi}_G^\mathbb{1}(q,x)$ denote the polynomial introduced in the source for the linear form $\mathbb{1}=(1,1,\ldots,1)$. Bajo et al.'s con…

Constant-playability relaxation. Let $G=(V,E)$ be a graph without isolated edges. Then there exists a constant $c\in\mathbb{N}\cup\{0\}$ such that $G$ is $f$-unbalanceable, where

For $r\in\mathbb{Z}^{+}$, let $a,b:\mathcal{P}([r])\to[0,1]$ be an optimal solution to the coloring problem, and let $a_i$ and $b_i$ denote the quantities associated with color…

For $r\in\mathbb{Z}^{+}$, let $a,b:\mathcal{P}([r])\to[0,1]$ be an optimal solution to the coloring problem, and write $a(R)$ and $b(R)$ for the corresponding aggregate quantities…

Song, Wei, Zhang, and Zhao's conjecture. For all $k\ge2$ and even $m\ge4$,

Zhao and Wei's conjecture. For all $k\ge2$ and $m\ge2$,

Mao, Wang, Magnant, and Schiermeyer's conjecture. For $k\ge2$,

Su and Liu's conjecture. $H$ is Ramsey-full if and only if $H$ is Gallai-Ramsey-full.

For $m\geq4$, let $\operatorname{ER}(m,4,r)$ denote the off-diagonal unordered Erdős–Rado number in the setting where monochromatic cliques have order $m$, lexical cliques have ord…

Let $\operatorname{ER}(4,4,r)$ denote the off-diagonal unordered Erdős–Rado number when monochromatic and lexical copies of $K_4$ are forbidden and a rainbow clique has order $r$.…

For integers $m,\ell\geq 4$, let $\operatorname{ER}(m,\ell,r)$ denote the off-diagonal unordered Erdős–Rado number for monochromatic and lexical cliques of the indicated orders aga…

Let $\operatorname{\mathcal{T}}_{4,n}$ denote the set of interlacing triangular arrays of rank $4$ and height $n$. Let $\square_n$ be the $n\times n$ square grid graph, and let…