2023/08/09 by Alkin, E., Dzhenzher, S., Nikitenko, O. +2
#05-01 #05C25 #05C65 #55-01 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #History and Overview (math.HO)
paper · doi:10.48550/arxiv.2308.05175
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.