34 problems

Hamiltonicity conjecture. The following four statements are equivalent, and each holds:

Let $G$ be a claw-free graph, and let $\chi_\ell(G)$ denote its list chromatic number and $\operatorname{\Delta_2}(G)$ its maximum codegree. The claw-free list-coloring conjecture.…

Let $G$ be a graph. It is claw-free if it has no induced subgraph isomorphic to a claw, and $G$ is 4-connected if deleting fewer than four vertices does not disconnect it. Matthews…

Let $G$ be a connected graph with chromatic number $t$. It is double-critical if for every edge $xy\in E(G)$, the graph $G\backslash\{x,y\}$ is $(t-2)$-colorable. The complete grap…

A graph is claw-free if it has no induced subgraph isomorphic to $K_{1,3}$. Let $\chi(G)$ and $\chi_\ell(G)$ denote the chromatic and list chromatic numbers. Gravier–Maffray's conj…

Matthews–Sumner conjecture. Every $4$-connected claw-free graph is hamiltonian.

Let $l\ge 1$ and $k\ge l$ be integers, and let $G$ be a connected claw-free $(k+l+1)$-regular graph of order $n$. Let $\gamma_{p,k}(G)$ denote the minimum cardinality of a $k$-powe…

Let $G$ be a claw-free subcubic graph. Seven-color packing coloring conjecture. The graph $G$ is $(2,2,2,2,2,2,2)$-packing colorable. The paper gives related packing-coloring resul…

Let $G$ be a graph. The parameters $Z(G)$ and $Z_+(G)$ denote the standard and positive semidefinite zero forcing numbers of $G$, respectively. A graph is claw-free if it has no in…

Let $G$ be a graph, and call the connected graph with degree sequence $3,3,3,1,1,1$ a net, with its vertices of degree $1$ called endvertices. Call $G$ claw-free if it has no induc…

Let $G$ be a finite claw-free graph, meaning that it has no induced subgraph isomorphic to $K_{1,3}$. Let $Z(w)$ be the hard-core lattice-gas partition function of $G$. Hamidoune–S…

Let $G$ be a claw-free graph, meaning that $G$ contains no induced copy of $K_{1,3}$. Let $\operatorname{dmg}(G)$ and $\operatorname{zdmg}(G)$ denote its ordinary and zombie damage…

A graph is claw-free perfect if it is both claw-free and perfect. Let $\chi(G)$ and $\chi_\ell(G)$ denote its chromatic and list chromatic numbers. Gravier et al.'s conjecture. For…

Let $G$ be a graph, and let $\Delta(G)$ denote its maximum degree. A clique of size $\Delta(G)$ is a complete subgraph on $\Delta(G)$ vertices. Cranston–Rabern's list-coloring conj…

Let $G$ be a claw-free subcubic graph, and let $\mathcal{G}_{11}$ be the graph specified in the paper. Exceptional packing coloring conjecture. If $G\neq\mathcal{G}_{11}$, then $G$…

Let $G$ be a graph. Write $\operatorname{\Delta_2}(G)$ for its maximum codegree and $\operatorname{\Delta}(G)$ for its maximum degree. Vu's conjecture. Fix…

Let $P_k$ denote the path on $k$ vertices, let $\operatorname{claw}=K_{1,3}$, and let $c(G)$ denote the cop number of a graph $G$. A graph is $(P_k,\operatorname{claw})$-free if it…

Let $G$ be a claw-free graph, meaning that $G$ has no induced subgraph isomorphic to $K_{1,3}$. Let $\delta(G)$ denote its minimum degree and let $\alpha(G)$ denote its independenc…

Let $G$ be a simple graph with $n$ vertices. A clique is a set of vertices inducing a complete graph, and the edge clique cover number $ecc(G)$ is the minimum number of cliques suc…

Claw-free Tutte-path conjecture. For every pair of vertices $a,b$ of a connected claw-free graph, there is a maximal $(a,b)$-path which is a Tutte path.

Jackson's conjecture. Every $2$-connected claw-free graph has a Tutte cycle.

Claw-free Berge conjecture. For every claw-free bridgeless cubic graph $G$,

Let $k$ be a non-negative integer and let $G$ be a connected claw-free graph of order $n$. For an integer $r\geq 2$, let $\sigma_r(G)$ be the minimum degree sum of an independent s…

De Joannis de Verclos–Kang–Pastor conjecture. For any claw-free graph $G$,

Let $G$ be a graph on $n$ vertices, and let $\alpha(G)$ denote its independence number, the maximum size of a set of pairwise nonadjacent vertices. The graph is triad-free when…