# Graph homomorphisms: structure and symmetry

@inproceedings{Hahn1997GraphHS, title={Graph homomorphisms: structure and symmetry}, author={Gena Hahn and Claude Tardif}, year={1997} }

This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex-transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a…

## 161 Citations

### The Homomorphism Structure of Classes of Graphs

- MathematicsCombinatorics, Probability and Computing
- 1999

We consider three aspects of homomorphisms of graphs and hypergraphs which are related to the structure of colour classes: (1) density, (2) the fractal property and (3) the generation of colour…

### Variations on a Theme: Graph Homomorphisms

- Mathematics
- 2013

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms. A core of a graph X is a vertex minimal subgraph to…

### Fractional Multiples of Graphs and the Density of Vertex-Transitive Graphs

- Mathematics
- 1999

We introduce a construction called the fractional multiple of a graph. This construction is used to settle a question raised by E. Welzl: We show that if G and H are vertex-transitive graphs such…

### GRAPH PARAMETERS VIA OPERATOR SYSTEMS

- Mathematics
- 2015

This work is an attempt to bridge the gap between the theory of operator systems and various aspects of graph theory. We start by showing that two graphs are isomorphic if and only if their…

### On No-Homomorphism Conditions

- Mathematics
- 2003

In this talk we review some general necessary conditions for the existence of graph homomorphisms [1, 4, 5]. Also we introduce some algebraic no-homomorphism theorems [1, 2, 3]. On the other hand, we…

### Graph homomorphisms through random walks

- MathematicsJ. Graph Theory
- 2003

In this paper, some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases are introduced and some information is obtained about the cores of vertex–transitive graphs.

### Automorphisms of graphs

- Mathematics
- 2003

This chapter surveys automorphisms of finite graphs, concentrating on the asymmetry of typical graphs, prescribing automorphism groups (as either permutation groups or abstract groups), and special…

## References

SHOWING 1-10 OF 131 REFERENCES

### Fractional Multiples of Graphs and the Density of Vertex-Transitive Graphs

- Mathematics
- 1999

We introduce a construction called the fractional multiple of a graph. This construction is used to settle a question raised by E. Welzl: We show that if G and H are vertex-transitive graphs such…

### Homomorphisms of graphs into odd cycles

- MathematicsJ. Graph Theory
- 1988

We give a class of graphs G for which there exists a homomorphism (= adjacency preserving map) from V(G) to V(C), where C is the shortest odd cycle in G, thereby extending a result of Albertson,…

### Path homomorphisms

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1996

Abstract We investigate homomorphisms between finite oriented paths. We demonstrate the surprising richness of this perhaps simplest case of homomorphism between graphs by proving the density theorem…

### Colorings and interpretations: a connection between graphs and grammar forms

- MathematicsDiscret. Appl. Math.
- 1981

### Homomorphisms to oriented cycles

- MathematicsComb.
- 1993

This work discusses the existence of homomorphisms to oriented cycles and gives a characterization of those digraphs that admit, a homomorphism toC, which can be used to prove the multiplicativity of a certain class of oriented cycles, and complete the characterization of multiplicative oriented cycles.

### Vertex-transitive graphs

- Mathematics
- 1964

. A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the…