8 problems

Let $G$ be a connected graph with at least three vertices. An edge weighting assigns a weight from $\{1,2,3\}$ to every edge of $G$, and the sum at a vertex is the sum of the weigh…

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 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 $d$-regular graph on $n$ vertices with $d>1$. An irregular edge weighting assigns real weights to the edges so that all vertices have distinct weighted degrees, and…

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…