# The Categories of Graphs

2011

Dissertation

## Degree Type

Doctor of Philosophy (PhD)

Mathematics

## Department or School/College

Department of Mathematical Sciences

George McRae

## Commitee Members

Peter Golubtsov, Kelly McKinnie, Nikolaus Vonessen, Joel Henry

## Keywords

Category of Graphs, Elementary Theory of the Categories of Graphs, Graph Morphisms, Categories of Graphs, Elementary Theory, Hedetniemi Conjecture

## Abstract

In traditional studies of graph theory, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct, and the graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these restricted morphisms as strict morphisms. We relax the conditions on the graphs by allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call the broader category of these graphs and these morphisms the Category of Conceptual Graphs and Graph Morphisms, denoted Grphs. We then define four other concrete categories of graphs created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs.

We determine the categorial structure of these six categories of graphs by characterizing common categorially defined structures and properties and by characterizing six special types of monomorphisms, and dually six special types of epimorphisms. We also establish the Fundamental Morphism Theorem in two of the categories of graphs.

We then provide an Elementary Theory for five categories of graphs, producing a list of first-order axioms that, when taken with the higher-order axiom of the existence of small products and coproducts, characterizes these five categories of graphs. We also provide a result toward Hedetniemi's conjecture that arose from the study of the categories of graphs.

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