Cycles in graphs and in hypergraphs: results and problems

TL;DR

This expository work surveys 1-cycles in graphs and 2-cycles in hypergraphs, with a focus on decomposition, bases, and representations in square-like constructions. It develops a Künneth-type framework showing that 2-cycles in the square $K^2$ are generated by products of 1-cycles from $K$, and extends these ideas to hypergraphs via tetrahedra as 2-cycles and to geometric interpretations using Cartesian products. The text provides explicit enumerations, constructs bases for cycle spaces, and discusses symmetric/coboundary phenomena, along with numerous open problems. Together, these results connect combinatorial cycle theory with topological and geometric viewpoints, enriching the understanding of cycle spaces in graphs and higher-dimensional analogues.

Abstract

This is an expository paper. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. E.g., a cycle in the sense of graph theory is a $1$-cycle, but not vice versa. It is easy to check that the sum (modulo $2$) of $1$-cycles is a $1$-cycle. In this text we study the following problems: to find $\bullet$ the number of all 1-cycles in a given graph; $\bullet$ a small number of 1-cycles in a given graph such that any 1-cycle is the sum of some of them. We also consider generalizations (of these problems) to graphs with symmetry, and to $2$-cycles in $2$-dimensional hypergraphs.

Figures (6)

• Figure 1: A $7$-vertices triangulation of the torus
• Figure 2: The cylinders over $K_2$, $K_3$, $K_{3,1}$, $K_5$
• Figure 3: A cycle of length 6 is the sum of three cycle of length 4
• Figure 4: Left: $K^{\square 2}_{2,1}$ is the $3\times 3$ grid graph. Middle: $K^{\square 2}_3$ on the torus $K^2_3$. Right: $K^{\square 2}_{3,1}$.
• Figure 5: Left: cuboctahedron; the union of edges is $K_4^{\square\underline2}$ (three of the six boundaries are highlighted); the complement to triangular faces is $K_4^{\underline2}$. Right: the same with some explanations; the figure does not show the invisible part whose projection is obtained from the pictured projection by rotation through $\pi/3$