Properties of Graph Doubling

Oct 13, 2025 by windailys.com 7 views

Introduction

In graph theory, the Hadwiger number of a graph $G$ , denoted by $\eta(G)$ , is the largest number $k$ such that $G$ contains a $k$ -clique. On the other hand, the chromatic number of a graph $G$ , denoted by $\chi(G)$ , is the minimum number of colors needed to color the vertices of $G$ such that no two adjacent vertices have the same color. In this article, we will discuss the problem of constructing a graph $2G$ such that $\chi(2G) = 2\chi(G)$ and $\eta(2G) = 2\eta(G)$ for every graph $G$ .

Background

The problem of constructing a graph $2G$ such that $\chi(2G) = 2\chi(G)$ and $\eta(2G) = 2\eta(G)$ is a well-known problem in graph theory. The problem is motivated by the fact that many graph properties are preserved under graph doubling. Graph doubling is a process of constructing a new graph $2G$ from a given graph $G$ by replacing each vertex of $G$ with a clique of size $k$ and each edge of $G$ with a complete bipartite graph $K_{k,k}$ .

Graph Doubling

Graph doubling is a process of constructing a new graph $2G$ from a given graph $G$ by replacing each vertex of $G$ with a clique of size $k$ and each edge of $G$ with a complete bipartite graph $K_{k,k}$ . The graph $2G$ is called the $k$ -fold graph of $G$ . The graph $2G$ has the following properties:

The chromatic number of $2G$ is at least $2\chi(G)$ .
The Hadwiger number of $2G$ is at least $2\eta(G)$ .

Double Coloring

The double coloring problem is the problem of constructing a graph $2G$ such that $\chi(2G) = 2\chi(G)$ . The double coloring problem is a well-known problem in graph theory. The problem is motivated by the fact that many graph properties are preserved under graph doubling.

Hadwiger Number

The Hadwiger number of a graph $G$ , denoted by $\eta(G)$ , is the largest number $k$ such that $G$ contains a $k$ -clique. The Hadwiger number is a well-known graph parameter in graph theory. The Hadwiger number is used to measure the size of the largest clique in a graph.

Properties of Graph Doubling

Graph doubling has several properties that make it a useful tool in graph theory. Some of the properties of graph doubling are:

Preservation of chromatic number: The chromatic number of $2G$ is at least $2\chi(G)$ .
Preservation of Hadwiger number: The Hadwiger number of $2G$ is at least $2\eta(G)$ .
Preservation of clique number: The clique number of $2G$ is at least $2\omega(G)$ .
Preservation of independence number: The independence number of $2G$ is at least $2\alpha(G)$ .

Constructions of Graph Doubling

There are several constructions of graph doubling that have been proposed in the literature. Some of the constructions of graph doubling are:

Clique replacement: Replace each vertex of $G$ with a clique of size $k$ .
Edge replacement: Replace each edge of $G$ with a complete bipartite graph $K_{k,k}$ .
Vertex replacement: Replace each vertex of $G$ with a complete graph $K_k$ .

Examples of Graph Doubling

There are several examples of graph doubling that have been proposed in the literature. Some of the examples of graph doubling are:

Complete graph: The complete graph $K_n$ is a graph with $n$ vertices and $n(n-1)/2$ edges. The graph $2K_n$ is a graph with $n^2$ vertices and $n^2(n-1)/2$ edges.
Cycle graph: The cycle graph $C_n$ is a graph with $n$ vertices and $n$ edges. The graph $2C_n$ is a graph with $n^2$ vertices and $n^2$ edges.
Complete bipartite graph: The complete bipartite graph $K_{m,n}$ is a graph with $m+n$ vertices and $mn$ edges. The graph $2K_{m,n}$ is a graph with $m^2+n^2$ vertices and $m^2n^2$ edges.

Open Problems

There are several open problems in graph doubling that remain to be solved. Some of the open problems in graph doubling are:

Double coloring problem: Is there a graph $2G$ such that $\chi(2G) = 2\chi(G)$ for every graph $G$ ?
Hadwiger number problem: Is there a graph $2G$ such that $\eta(2G) = 2\eta(G)$ for every graph $G$ ?
Clique number problem: Is there a graph $2G$ such that $\omega(2G) = 2\omega(G)$ for every graph $G$ ?

Conclusion

Q: What is graph doubling?

A: Graph doubling is a process of constructing a new graph $2G$ from a given graph $G$ by replacing each vertex of $G$ with a clique of size $k$ and each edge of $G$ with a complete bipartite graph $K_{k,k}$ .

Q: What are the properties of graph doubling?

A: Graph doubling has several properties that make it a useful tool in graph theory. Some of the properties of graph doubling are:

Preservation of chromatic number: The chromatic number of $2G$ is at least $2\chi(G)$ .
Preservation of Hadwiger number: The Hadwiger number of $2G$ is at least $2\eta(G)$ .
Preservation of clique number: The clique number of $2G$ is at least $2\omega(G)$ .
Preservation of independence number: The independence number of $2G$ is at least $2\alpha(G)$ .

Q: What are the constructions of graph doubling?

A: There are several constructions of graph doubling that have been proposed in the literature. Some of the constructions of graph doubling are:

Clique replacement: Replace each vertex of $G$ with a clique of size $k$ .
Edge replacement: Replace each edge of $G$ with a complete bipartite graph $K_{k,k}$ .
Vertex replacement: Replace each vertex of $G$ with a complete graph $K_k$ .

Q: What are the examples of graph doubling?

A: There are several examples of graph doubling that have been proposed in the literature. Some of the examples of graph doubling are:

Complete graph: The complete graph $K_n$ is a graph with $n$ vertices and $n(n-1)/2$ edges. The graph $2K_n$ is a graph with $n^2$ vertices and $n^2(n-1)/2$ edges.
Cycle graph: The cycle graph $C_n$ is a graph with $n$ vertices and $n$ edges. The graph $2C_n$ is a graph with $n^2$ vertices and $n^2$ edges.
Complete bipartite graph: The complete bipartite graph $K_{m,n}$ is a graph with $m+n$ vertices and $mn$ edges. The graph $2K_{m,n}$ is a graph with $m^2+n^2$ vertices and $m^2n^2$ edges.

Q: What are the open problems in graph doubling?

A: There are several open problems in graph doubling that remain to be solved. Some of the open problems in graph doubling are:

Double coloring problem: Is there a graph $2G$ such that $\chi(2G) = 2\chi(G)$ for every graph $G$ ?
Hadwiger number problem: Is there a graph $2G$ such that $\eta(2G) = 2\eta(G)$ for every graph $G$ ?
Clique number problem: Is there a graph $2G$ such that $\omega(2G) = 2\omega(G)$ for every graph $G$ ?

Q: What are the applications of graph doubling?

A: Graph doubling has several applications in graph theory and computer science. Some of the applications of graph doubling are:

Graph coloring: Graph doubling can be used to construct graphs with specific chromatic numbers.
Hadwiger number: Graph doubling can be used to construct graphs with specific Hadwiger numbers.
Clique number: Graph doubling can be used to construct graphs with specific clique numbers.

Q: What are the challenges in graph doubling?

A: Graph doubling has several challenges that need to be addressed. Some of the challenges in graph doubling are:

Computational complexity: Graph doubling can be computationally expensive, especially for large graphs.
Graph structure: Graph doubling can be sensitive to the structure of the input graph.
Graph properties: Graph doubling can affect the properties of the input graph.

Q: What are the future directions in graph doubling?

A: There are several future directions in graph doubling that need to be explored. Some of the future directions in graph doubling are:

Developing new constructions: Developing new constructions of graph doubling that can handle large graphs.
Improving computational efficiency: Improving the computational efficiency of graph doubling algorithms.
Exploring new applications: Exploring new applications of graph doubling in graph theory and computer science.