2017/11/01 by Artur Gizycki, Gizycki, Artur
Computer Science · Mathematics · #05C40 (Secondary) #26E25 (Primary) #97E60 #Advanced Graph Theory Research #FOS: Mathematics #General Mathematics (math.GM) #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.1711.00330
openalex publication_date 2017/11/01 · openalex created_date 2017/11/10 · openalex updated_date 2026/07/28
We present the theory of multifunctions applied to graphs. Its interesting feature is that walks are recognized as iterations. We consider the graphs with arbitrary number of vertices which are determined by multifunctions. The mutually unique correspondence between graphs and multifunctions is proven. We explain that many facts of graph theory can be formulated in the language of multifunctions and as examples we give\colon neighborhood, walk, independent set, clique, bipartiteness, connectedness, isolated vertices, graph metric, leaf. To simplify the proofs of our theorems, we introduce the concept of iterations of multifunctions. The new equivalent conditions for bipartite multifunctions including the König theorem and even iterations theorem are given. We prove that there exist filters and ideals in graph theory that are similar to those from the set theory. Finally, to illustrate these facts, we consider the multifunction of prime numbers.