Question

Given a self-complementary graph with \(4 m\) vertices for some \(m>0\), construct a self-complementary graph on \(4 m+1\) vertices that contains the self- complementary graph on \(4 m\) vertices as an induced subgraph.

Short answer

Add one vertex; connect it to half of the existing vertices.

Step-by-step solution

Understanding Self-Complementary Graphs

A graph is self-complementary if it is isomorphic to its complement. For a graph with \( n \) vertices to be self-complementary, the equation \( inom{n}{2} = 2k \) (where \( k \) is the number of edges) must hold, implying the necessity for the vertex count to satisfy \( n(n-1)/4 \) to be an integer. This holds for \( n = 0, 1, 4, 5, ext{ and multiples of} \, 4m \).

Given Condition and Purpose

We are given a self-complementary graph with \( 4m \) vertices and need to construct another self-complementary graph with \( 4m+1 \) vertices that includes the initial \( 4m \) vertices graph as an induced subgraph.

Constructing the New Vertex

Add one new vertex (vertex \( v \)) to the graph on \( 4m \) vertices. Initially, connect this vertex \( v \) to half of the original \( 4m \) vertices. This can always be done since \( 2m \) is an integer.

Completing the Self-Complementarity

After adding the one vertex and connecting it to half of the \( 4m \) original vertices, check the complement graph. You should reassign connections based on the complement in the same manner (add the connections that are missing in the complement from \( v \) and remove the existing ones). This symmetry maintains the property required for self-complementarity.

Verify Self-Complementarity

Double-check that both the graph and its complement are indeed isomorphic by verifying that rearranging the vertices maintains identical connection attributes reflected in the complement.

Key concepts

Graph Theory

Graph theory is a fascinating area of mathematics and computer science. It explores the properties and connections between entities called "vertices" (or "nodes") and the relationships between these vertices known as "edges." Graphs are used to model and analyze networks in various fields such as social sciences, biology, and computer networking.

In graph theory, different types of graphs are studied like directed, undirected, weighted, and unweighted graphs. Each type presents unique characteristics and applications, like analyzing social networks or computer circuits.

Some fundamental concepts in graph theory include:

• Paths: A sequence of vertices connected by edges.
• Cycles: A path where the start and end vertices are the same.
• Connected Graphs: A graph in which every vertex is linked directly or indirectly.
• Complete Graphs: Graphs where every pair of vertices is directly connected by an edge.

This foundational knowledge is key to understanding more complex graph-related concepts, such as induced subgraphs and isomorphic graphs.

Induced Subgraph

An induced subgraph is a crucial concept in graph theory, especially when discussing self-complementary graphs. It is essentially a smaller graph obtained from a larger graph by selecting a subset of its vertices and including all the edges between them that were present in the larger graph.

Imagine you have a network of friends on a social media platform. If you choose a group of specific friends and consider only their direct connections, you've created an induced subgraph of the overall network.

The use of induced subgraphs helps in understanding and solving complex problems by dividing them into smaller, more manageable sections. In the given problem, the graph with \(4m\) vertices is part of a larger graph with \(4m+1\) vertices, where the original graph remains unchanged in terms of connections among the \(4m\) vertices. This way, analyzing the overall structure becomes simpler as one can focus on each section independently.

Isomorphic Graphs

Graph isomorphism is a concept where two graphs are considered equivalent if their structures can be mapped onto each other in a certain way. Specifically, two graphs are isomorphic if there's a one-to-one correspondence between their vertex sets and edge sets, maintaining the connection structure.

Consider isomorphic graphs as twin puzzles – even if pieces are rearranged, the arrangement stays the same. Observing their patterns, connections, and adjacency lists remains consistent no matter how they are flipped or rotated.

• Vertex Matching: Every vertex in one graph matches with a vertex in the other.
• Edge Correspondence: Every edge in one graph has a corresponding edge in the other.

Understanding this is vital when discussing self-complementary graphs. A self-complementary graph is isomorphic to its complement, meaning the graph and its complement can be matched perfectly.

This concept is crucial for verifying and constructing graphs in mathematical problems, ensuring that transformations maintain the graph's inherent structure.