15 problems

Cambie–Kang conjecture. For every $\varepsilon>0$, there is $d_0\in\mathbb{N}$ such that, if $H$ has maximum degree $d\geqslant d_0$ and

Let $(G,\sigma)$ be a planar signed graph. A signed DP-coloring is a correspondence coloring for signed graphs, with admissible options assigned to vertices as in Jin, Kang, and St…

Let $k\geq 1$ and let $G$ be a graph with $\Delta(G)\leq k$, where $\Delta(G)$ is the maximum degree. Every such graph is DP $(k+1)$-colorable and, by the Hajnal–Szemerédi theorem,…

Let $n$ be a positive integer and let $k\geq n+1$. The complete bipartite graph $K_{n,n}$ is DP $k$-colorable and equitably $k$-colorable. In the source, SEL $k$-colorable denotes…

DP-4-colorability conjecture. Every planar graph without chorded $6$-cycles is DP-$4$-colorable.

Let $G$ be a graph on $n$ vertices, and let $P^{*}_{DP}(G,k)$ denote its dual DP color function, the maximum number of colorings over all full $k$-fold covers of $G$. Dual-DP Shame…

Counting conjecture. For all sufficiently large $t$ and every $(k+t)$-fold cover $\mathcal{H}$ of $G\vee K_t$, the number of proper $\mathcal{H}$-colorings of $G\vee K_t$ is at lea…

Let $G$ be a triangle-free planar graph with $n$ vertices, and let $P_{DP}(G,4)$ denote its DP color function. The exponential DP-coloring conjecture. There exists a constant $c>1$…

Let $k$ be a power of a prime. For a graph $G$, let $P_{\mathcal{L}_k}(G,k)$ be the minimum number of colorings over all $\mathcal{L}_k$-labelings, where $\mathcal{L}_k$ is the col…

DP-version of the Alon–Krivelevich–Sudakov conjecture. For every graph $F$, there exist constants $c,d_0>0$ such that, whenever $H$ is $F$-free, has maximum degree $d\geqslant d_0$…

Let $G$ be a graph and let $K$ be a clique in $G$ with $|K|\geq 3$. For an $m$-fold cover $\mathcal{H}=(L,H)$ of $G$, say that it is conducive to $K$ if it is full and its restrict…

DP-coloring extension of the Alon–Krivelevich–Sudakov conjecture. For every graph $F$, there is a constant $c_F>0$ such that, if $H$ is $F$-free, has maximum degree $d\geqslant 2$,…

Let $G$ be a graph with odd girth, and let $P_{DP}(G,m)$ and $P(G,m)$ denote its DP color function and chromatic polynomial, respectively. Odd-girth eventual equality conjecture. T…

Unbounded excess conjecture. The difference

Let $G$ be a loopless multigraph, let $\Delta(G)$ denote its maximum degree, and let $\mathsf{Line}(G)$ be the simple graph whose vertices are the edges of $G$, with adjacency when…