First, prove that any two multigraphs G of order 3 with...

First, prove that any two multigraphs $G$ of order 3 with degree sequence $(4,4,4)$ are isomorphic. Then (a) Determine all the nonisomorphic induced subgraphs of $G$. (b) Determine all the nonisomorphic spanning subgraphs of $G$. (b) Determine all the nonisomorphic subgraphs of order 3 of $G$.

First, prove that any two multigraphs $G$ of order 3 with degree sequence $(4,4,4)$ are isomorphic. Then
(a) Determine all the nonisomorphic induced subgraphs of $G$.
(b) Determine all the nonisomorphic spanning subgraphs of $G$.
(b) Determine all the nonisomorphic subgraphs of order 3 of $G$.

Key Concepts

Graph Isomorphism

Graph isomorphism involves establishing a one-to-one correspondence between the vertex sets of two graphs that preserves the edge structure. It is a key concept in graph theory that allows us to determine when two graphs have essentially the same structure, even if their representations differ. In this context, proving that two multigraphs with the same degree sequence are isomorphic means showing that there exists a vertex mapping that aligns the connectivity and multiplicities in edges between the graphs.

Multigraph

A multigraph is a type of graph that permits multiple edges between the same pair of vertices. Unlike simple graphs, looped edges and parallel edges can be present, influencing the degree counts of vertices. This concept is critical here because the structure of a multigraph with a given degree sequence might uniquely determine the graph up to isomorphism, as is the case with the order 3, degree sequence (4,4,4) graphs.

Degree Sequence

The degree sequence of a graph is a list of its vertex degrees, usually arranged in non-increasing order. It serves as an important invariant in graph theory for comparing and classifying graphs. In this problem, the degree sequence (4,4,4) across all vertices imposes strict constraints on the possible graph configurations, thereby aiding in the isomorphism proof and in the classification of the subgraphs.

Induced Subgraph

An induced subgraph is formed by selecting a subset of vertices from the original graph along with all the edges that exist between those vertices in the original graph. This concept is crucial when determining nonisomorphic induced subgraphs since every induced subgraph retains the complete interaction among the chosen vertices, reflecting the inherent structure of the parent graph.

Spanning Subgraph

A spanning subgraph is a subgraph that includes all the vertices of the original graph, though it might not include every edge. This concept is important for examining how the removal of certain edges, while keeping the vertex set complete, creates different structural configurations that are nonisomorphic from one another.

General Subgraph

A subgraph of a graph is defined as any graph whose vertex set is a subset of the original graph's vertices and whose edge set is a subset of the original graph's edges. In the context of this problem, understanding general subgraphs, especially those of a specific order (in this case, 3), is fundamental for cataloguing all distinct structural variations up to isomorphism.

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