11 problems

Let $G=(V,E)$ be a nice graph, meaning a graph without isolated edges. A graph is $(k,k')$-choosable if every assignment of a $k$-element list of real numbers to each vertex and a…

Let $G=(V,E)$ be a nice graph, meaning a graph without isolated edges. A graph is edge-weight $3$-choosable if, for every assignment $L$ of a three-element set of real numbers to e…

Let $G$ be a nice graph, meaning a graph without isolated edges, and let $a,b,c$ be pairwise distinct real numbers. A proper edge weighting is an edge weighting for which adjacent…

Let $G=(V,E)$ be a simple graph. A proper total weighting is a map $w:V\cup E\to\mathbb R$ such that, for every edge $uv\in E$, the weighted degrees … A graph is $(2,2)$-choosable…

Let $G$ be a graph, and let an $(a,b)$-matrix mean a square matrix $A_G(\eta)$ in which each vertex column occurs at most $a$ times and each edge column occurs at most $b$ times. S…

Let $G$ be a graph, and define the graph matrix $A_G$ and the matrices $A_G(\eta)$ as above. An $(a,b)$-matrix is a square matrix $A_G(\eta)$ with $\eta(v)\le a$ for every vertex a…

Let $G$ be a graph, and orient its edges arbitrarily. Define the graph matrix $A_G$ with rows indexed by $E(G)$ and columns indexed by $V(G)\cup E(G)$ by assigning to an oriented e…

Let $G$ be a graph, and let $L$ assign to every vertex and edge $z\in V(G)\cup E(G)$ a set $L(z)$ of permissible real weights. An $L$-total weighting is a proper total weighting…

Let $G$ be a graph. A proper total weighting is a mapping $\phi:V(G)\cup E(G)\to\mathbb{R}$ such that, for every edge $uv\in E(G)$, … where $E(v)$ denotes the set of edges incident…

Let $G$ be a graph without isolated edges. A proper edge weighting is a mapping $\phi:E(G)\to\mathbb{R}$ such that, for every edge $uv\in E(G)$, the sums of the weights on edges in…

Irregularity strength conjecture. For every graph $G$, the value of $s(G)$ is equal to the lower bound $L(G)$ plus a constant independent of $G$.