7 problems

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…

Hajós' conjecture. Every simple Eulerian graph $G$ has a cycle decomposition with at most

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…

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.

Häggkvist–Kriesell conjecture. For every positive integer $\ell$, there is an integer $d_\ell$ such that every Eulerian graph $G$ with minimum degree at least $d_\ell$ admits an…