10 problems

Let a saturated graph be a regular Eulerian graph whose avoidance index equals its degree, and let a circulant graph be a graph defined by a generating set on a cyclic vertex group…

For a graph $G$, let $\operatorname{Maj}'(G)$ denote the least number of colors in a strong majority edge-coloring, meaning an edge-coloring $c:E(G)\to C$ such that for every edge…

Let $G$ be an Eulerian graph, and call an integer feasible if $G$ has a cycle decomposition containing that many odd cycles. Parity interval conjecture. If $r$ and $t$ are feasible…

Let $G$ be a finite graph and let $H$ be a subgraph of $G$ with at most $k$ components, where $k$ is a natural number. The notions $S$-Eulerian and semi $S$-Eulerian mean, respecti…

Let $G$ be an Eulerian graph, let $\Delta(G)$ be its maximum degree, and let a path odd-cover (respectively, cycle odd-cover) be a collection of paths (respectively, cycles) whose…

Let $G$ be a graph, and define $\nu(G)$ as the absolute value of $A_D(-1)$ for any orientation $D$ of $G$. An Eulerian graph is a graph in which every vertex has even degree. Euler…

Let $K^*_{2r+1,2r+1}$ be the complete bipartite graph $K_{2r+1,2r+1}$ with a perfect matching removed, and let $\operatorname{av}(G)$ denote the avoidance index of an Eulerian grap…

The open Eulerianity conjecture. A Peano continuum $X$ is open Eulerian if and only if all but two vertices of $X_\sim$ have even degree.

The Eulerianity conjecture. A Peano continuum $X$ is Eulerian if and only if every edge cut of $X$ is even.

Let $G$ be a $2d$-edge-connected, $2d$-regular graph of odd order, where $d\ge 1$. A rooted decomposition is a decomposition into odd closed trails that all share a common vertex,…