Iterations of Multifunctions for Graph Theory: Bipartite Graphs and Filters

TL;DR

This paper develops a multifunction framework for graph theory, representing walks as iterations, and provides new characterizations for bipartite graphs and related concepts, connecting graph properties with multifunction iterations.

Contribution

It introduces a novel multifunction approach to graph theory, establishing a correspondence between graphs and multifunctions and deriving new conditions for bipartiteness and other properties.

Findings

Established a one-to-one correspondence between graphs and multifunctions.

Derived new conditions for bipartite multifunctions, including a version of K"onig's theorem.

Connected graph theoretical concepts with filters and ideals from set theory.

Abstract

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 K\"onig 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.