13 problems

For positive integers $n$ and $k$ with $n\geq 2k$, let $\operatorname{KG}(n,k)$ be the Kneser graph and let $\chi_c(G)$ denote the circular chromatic number of a graph $G$. Johnson…

Circular chromatic index conjecture. The graph $G$ satisfies

Let $(G,-)$ be a signed graph whose underlying graph is $G$, with every edge negative, and let $(K_{k+1},-)$ denote the complete graph on $k+1$ vertices with every edge negative. A…

Circular-colouring density conjecture. If $G$ has no $(p,q)$-colouring, then there exist positive rational numbers $\varepsilon_{p,q}$ and $c_{p,q}$, depending on $p$ and $q$, such…

Chang–Huang–Zhu conjecture. Whenever $n\geq d+2$,

DP-completeness conjecture. The problem $\mathrm{CCHI}^{=_r}$ is DP-complete.

Let $G$ and $H$ be graphs, let $G\times H$ be their categorical product, and let $\chi_c(G)$ denote the circular chromatic number. Zhu's circular chromatic conjecture. For any grap…

Let $(G,\sigma)$ and $(H,\tau)$ be signed graphs, and let $(G,\sigma) \times (H,\tau)$ denote their tensor (direct) product, whose vertices $(u,x)$ and $(v,y)$ are adjacent when…

Let $(G,\sigma)$ be a signed bipartite planar graph, let $S(K_3)$ be the signed subdivision of $K_3$, and write $(G,\sigma)\to(K_{3,3},M)$ for the corresponding signed homomorphism…

Let $p$ be a positive integer, and let $\mathcal{P}^*_{4p}$ denote the class of signed bipartite planar graphs with the indicated girth condition. Bipartite analogue of the Jaeger–…

Let $p\ge 5$ be a prime, and let $f(p)$ be the smallest number such that every planar graph of girth $p$ with no cycles of lengths from $p+1$ through $f(p)$ admits a homomorphism t…

Let $t$ be a positive integer and let $G$ be a planar graph of girth at least $2t$. The circular chromatic conjecture. … The cases $t=1$ and $t=2$ follow from the Four Color Theore…

Let ${\mathcal S}'=\left\{\chi'_c(G)\,\middle|\,G\text{ is a graph}\right\}$ be the set of circular edge chromatic numbers of graphs. Zhu's accumulation-point conjecture. Integers…