In [Structural properties and decomposition of linear balanced matrices, Mathematical Programming, 55:129–168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for bal...

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, implies the existence of the (non-homogeneous) cd-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian po...

A group-labeled graph is a graph whose vertices and edges have been assigned labels from some abelian group. The weight of a subgraph of a group-labeled graph is the sum of the labels of the vertices and edges in the subgraph. A group-labeled graph is said to be balanced if the weight of every cycle in the graph is zero. We give a characterization of balanced group-labeled graphs that generaliz...

Let $$n\geqslant 1$$ be a number. $${\Gamma }_n$$ the graph on $$\mathbb {R}^n$$ connecting points of rational Euclidean distance. It is consistent with choiceless set theory $$\textrm{ZF}\,{+}\,\textrm{DC}$$ that has countable chromatic number, yet number $$\Gamma _{n+1}$$ uncountable.