29 problems

Let $d$ be sufficiently large, and let $G$ be a $d$-degenerate triangle-free graph with fractional chromatic number $\chi_f(G)$. Martinsson–Steiner conjecture. The following two as…

Let $H$ be an $r$-uniform $d$-degenerate triangle-free hypergraph, with $r\geq 3$. Here, $\chi_f(H)$ denotes the fractional chromatic number, and $c_r>0$ is a constant depending on…

Let $G$ be a graph, let $\chi(G)$ and $\omega(G)$ denote its chromatic and clique numbers, let $\chi_f(G)$ denote its fractional chromatic number, and let $H$ be a largest induced…

Let $W_t$ be the join of the cycle $C_t$ with $K_1$, let $\alpha(G)$ be the independence number, let $\chi_f(G)$ be the fractional chromatic number, and let $\Box$ denote the Carte…

Let $W_t$ be the join of the cycle $C_t$ with $K_1$, let $\chi_f(G)$ denote the fractional chromatic number of a graph $G$, and let $W_{2t+1}^{\Box 2}$ be the Cartesian square of t…

Let $G$ be a planar graph, and let $a_f(G)$ denote its fractional arboricity: the minimum ratio $p/q$ for which vertices can be assigned at least $q$ colors from a total of at most…

Let $n$ tend to infinity, and let $G$ be a triangle-free graph on $n$ vertices. Write $\chi_f(G)$ for its fractional chromatic number. Cames van Batenburg–de Joannis de Verclos–Kan…

Let $G$ be a triangle-free graph, and let $d_G(v)$ denote the degree of a vertex $v\in V(G)$. Consider a probability distribution on the independent sets of $G$. Kelly–Postle's loc…

Let $\mathcal{L}=\{\mathcal{M}_1,\ldots,\mathcal{M}_k\}\in\mathcal{D}^k$ be matroids on $V$, let $\mathcal{C}=\bigcap\mathcal{L}$, and let $\vec h\in\mathbb{R}_{\ge0}^V$. Weighted…

Let $G$ be a 4-cycle-free planar graph, and let $\chi_f(G)$ denote its fractional chromatic number. Fractional chromatic conjecture. There exists some $\epsilon>0$ such that … The…

Intermediate fractional colorability conjecture. Every such graph $G$ is fractionally $19/7$-colorable.

Heckman–Thomas planar conjecture. Every subcubic triangle-free planar graph is fractionally $8/3$-colorable.

Let $G$ be a triangle-free subcubic graph, and let $F_{11}$, $F_{14}^{(1)}$, $F_{14}^{(2)}$, $F_{19}^{(1)}$, $F_{19}^{(2)}$, and $F_{22}$ be the six specified graphs. Let…

Multiplicative fractional list-coloring conjecture. If $G$ is $(f,N)$-list-colorable, then $G$ is $(f,mN)$-list-colorable for every positive integer $m$.

Let $G$ be a graph, let $\chi_f(G)$ denote its fractional chromatic number, and let $\rho(G)$ denote its Hall ratio, the maximum of $v(H)/\alpha(H)$ over all non-null subgraphs…

Let $G$ be a planar graph of girth at least five, and let $\chi_f(G)$ denote its fractional chromatic number. Dvořák–Mnich conjecture. There exists a real number $c<3$ such that ev…

A graph $G$ is $4$-regular if every vertex has degree $4$. For a graph $G$, let $D(G,2)$ denote the minimum defect in a fractional $2$-coloring. The exceptional graphs are the comp…

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…

A $2$-coloring assigns each vertex a red usage in $[0,1]$, with blue usage $1-r(v)$; the defect of a vertex is the sum, over its neighbors, of the overlap in their color usages, an…

For a graph $Q$ and $r\in\mathbb{N}\cup\{\infty\}$, let $\mathcal{C}_r(Q)$ consist of the maximal cliques of size less than $r$ together with the cliques of size exactly $r$, and d…

Let $G$ be a triangle-free graph that is $d$-degenerate, meaning every subgraph of $G$ has a vertex of degree at most $d$. The fractional chromatic bound. Then … The claim follows…

The bounded fractional-chromatic-number-to-Hall-ratio conjecture. There is an absolute constant such that, for every graph $G$,

Girth-five fractional-coloring conjecture. There exists $\delta>0$ such that every planar graph of girth at least five has fractional chromatic number at most

Let $G$ be a graph with maximum degree $5$ and clique number $4$, such that no two $4$-cliques intersect and no vertex outside any maximum clique $C$ has more than one neighbour in…

For each $\Delta\geq 3$, define $f(\Delta)$ as the best universal fractional-coloring deficit for graphs of maximum degree $\Delta$ in the paper's setting. Monotonicity conjecture.…