10.4 Special types of graphs (bipartite, complete, regular)
3 min read•july 30, 2024
Graph theory gets even cooler when we look at special types of graphs. Bipartite graphs split vertices into two groups, complete graphs connect everything, and regular graphs have uniform degrees. These structures pop up everywhere!
Understanding these special graphs is key to solving real-world problems. They model everything from job assignments to social networks. Plus, they're the building blocks for more complex graph concepts we'll tackle later.
Bipartite graphs
Definition and characteristics
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- divides vertices into two disjoint sets with edges connecting vertices between sets
- Two disjoint sets called partite sets or color classes
- Graph bipartite if and only if it contains no
- Km,n connects every vertex in one set to every vertex in other set
- Recognize bipartite graphs using assigning different colors to adjacent vertices
- Bipartite graphs precisely the graphs colored using only two colors
Applications and related concepts
- Used in (assigning jobs to workers, students to dormitories)
- Closely related to concept
- Model relationships between two distinct groups (buyers and sellers, actors and movies)
- Represent alternating structures (chemical compounds, scheduling problems)
- Analyze social networks (identifying communities, studying interactions between groups)
- Optimize resource allocation (assigning tasks to processors, matching supply with demand)
Complete graphs
Properties and structure
- connecting every pair of distinct vertices with unique edge
- Denoted as Kn for n vertices
- Contains edges
- Every vertex has degree
- Maximally connected with highest possible number of edges for n vertices
- an empty graph (no edges)
- equal to number of vertices
Applications and significance
- Model all-to-all communication networks
- Solve Traveling Salesman Problem using heuristics
- Analyze worst-case scenarios in algorithm complexity
- Study social networks (complete subgraphs represent tightly connected groups)
- Crucial in extremal graph theory and Ramsey theory
- Benchmark for testing graph algorithms
- Represent fully connected neural networks in machine learning
Regular graphs
Definition and types
- Graph where every vertex has same degree
- has each vertex with degree k
- denoted by r (common degree of all vertices)
- Kn example of (n-1)-
- Q3 example of 3-regular graph (cubic graph)
- Regular graphs of degree 2 precisely disjoint unions of cycles
- Handshaking lemma implies number of vertices in odd degree regular graph must be even
Properties and examples
- Vertex-transitive (all vertices "look the same")
- Often exhibit high symmetry and interesting structural properties
- famous example of 3-regular graph with 10 vertices
- Platonic solids correspond to regular graphs (tetrahedron, cube, octahedron, dodecahedron, icosahedron)
- Complete bipartite graph Kn,n example of n-regular graph
- special class with additional constraints on neighbor connections
- smallest regular graphs with given degree and girth
Degree sequences vs Graph types
Degree sequences and graph properties
- lists degrees of vertices, typically in non-increasing order
- Bipartite graphs have constraints on degree sequences related to sum of degrees in each partite set
- Complete graphs have uniform degree sequence (all entries equal to n-1)
- Regular graphs have constant degree sequence (all entries equal to degree of regularity r)
- provides necessary and sufficient conditions for sequence to be degree sequence of simple graph
- constructs graph with given degree sequence or determines if such graph exists
- Degree sequence provides insights into graph properties (connectivity, planarity, existence of Hamiltonian cycles)
Analyzing and constructing graphs from degree sequences
- satisfy specific conditions (sum of degrees even, largest degree less than number of vertices)
- compares sequences to determine potential graph structures
- used to classify and analyze graph families
- characterized by special properties of their degree sequences
- attempts to determine unique graphs from degree sequences
- Degree sequence analysis helps in studying random graph models and their properties
- Degree sequence constraints used in graph generation algorithms and network modeling
Key Terms to Review (28)
Bipartite Graph: A bipartite graph is a type of graph where the set of vertices can be divided into two distinct sets such that no two vertices within the same set are adjacent. This structure allows for various applications, including matching problems and network flows, by facilitating the understanding of relationships between two different types of entities. It is particularly useful in problems related to scheduling, resource allocation, and colorings in graphs.
Bipartite matching algorithm: A bipartite matching algorithm is a method used to find the maximum matching in a bipartite graph, where vertices can be divided into two distinct sets such that no two graph vertices within the same set are adjacent. This algorithm helps in efficiently pairing elements from the two sets, which is crucial for solving problems like job assignments, network flows, and resource allocation.
Cage Graphs: Cage graphs are a special type of graph characterized by their high degree of symmetry and specific properties, such as having the minimum number of edges for a given number of vertices and a maximum degree. These graphs are often used in combinatorial design and can be connected to concepts like bipartite, complete, and regular graphs, as they explore the relationships between vertices while adhering to strict constraints.
Clique number: The clique number of a graph is defined as the size of the largest complete subgraph contained within it. This concept is essential for understanding the structure of graphs, particularly in identifying groups of vertices that are all directly connected to each other. The clique number provides insights into the connectivity and clustering properties of graphs, which can relate to various types such as bipartite, complete, and regular graphs.
Complement Graph: A complement graph is formed by taking a graph and connecting all the vertices that are not already connected by an edge in the original graph. This means that if there is an edge between two vertices in the original graph, there will be no edge between them in the complement graph, and vice versa. The complement graph provides insights into the relationships between vertices that are not directly connected, which is useful when analyzing special types of graphs like bipartite, complete, and regular graphs.
Complete bipartite graph: A complete bipartite graph is a special type of graph that consists of two disjoint sets of vertices, where every vertex in one set is connected to every vertex in the other set. This structure ensures that there are no edges connecting vertices within the same set, making it a specific form of bipartite graph. Complete bipartite graphs are often denoted as $K_{m,n}$, where $m$ and $n$ represent the number of vertices in each of the two sets. Their unique structure allows for important connections in various areas of combinatorics, such as graph theory and Ramsey theory.
Complete Graph: A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. This means that if there are 'n' vertices in the graph, there are exactly $$\frac{n(n-1)}{2}$$ edges. Complete graphs are fundamental in various areas of study, showcasing the concept of connectivity and serving as a base case for many combinatorial and graphical theories.
Cube Graph: A cube graph is a type of graph that represents the structure of a cube, where each vertex corresponds to a corner of the cube and each edge represents a connection between two corners. This graph is a special case of a hypercube graph, specifically the 3-dimensional hypercube, and exhibits properties such as symmetry and regularity, making it an important concept in understanding various special types of graphs.
Degree of regularity: The degree of regularity in graph theory refers to a specific property that characterizes the uniformity of a graph's structure, particularly in regular graphs. In such graphs, all vertices have the same degree, which means that each vertex connects to the same number of edges. This concept is crucial when examining special types of graphs, as it influences the overall connectivity and symmetry, impacting various properties like bipartiteness and completeness.
Degree Sequence: A degree sequence is a list of the degrees of the vertices in a graph, usually arranged in non-increasing order. It provides a way to summarize the connectivity of a graph and is crucial for understanding various properties, such as whether a particular graph can exist or its structure. The degree sequence helps in applying key concepts like the Handshaking Lemma, which relates the sum of the degrees to the number of edges, and in identifying special types of graphs such as bipartite, complete, and regular graphs.
Degree Sequence Majorization: Degree sequence majorization is a concept in graph theory that involves comparing two degree sequences, which are lists of the degrees of the vertices in a graph. It indicates whether one degree sequence can be transformed into another through a series of specific rearrangements, reflecting the structural properties of graphs and their vertices. Understanding degree sequence majorization is important in studying special types of graphs, as it helps determine their regularity, completeness, and bipartite nature.
Degree Sequence Partitions: Degree sequence partitions refer to the division of a graph's vertex degrees into distinct groups based on specific criteria or characteristics, allowing for the analysis of graph structures and properties. Understanding these partitions is essential in identifying special types of graphs, such as bipartite, complete, and regular graphs, as these classifications are inherently linked to how degree sequences can be structured or organized. By examining the degree sequence partitions, one can gain insight into the relationships between vertices and the overall topology of the graph.
Degree Sequence Reconstruction Problem: The degree sequence reconstruction problem is a challenge in graph theory that involves determining whether a given sequence of non-negative integers can represent the degree sequence of a simple graph. This problem is closely tied to understanding the structure and properties of different types of graphs, such as bipartite graphs, complete graphs, and regular graphs, as these structures often have specific constraints and characteristics related to their degree sequences.
Dfs for bipartite checking: DFS for bipartite checking refers to the use of Depth-First Search (DFS) to determine whether a graph can be colored using two colors without adjacent vertices sharing the same color. This process helps to identify if a graph is bipartite, which is an important characteristic in understanding its structure and properties, especially in relation to matching problems and graph algorithms.
Erdős-Gallai Theorem: The Erdős-Gallai Theorem is a fundamental result in graph theory that characterizes the degree sequences of simple graphs. Specifically, it states that a finite sequence of non-negative integers can be the degree sequence of a simple graph if and only if it satisfies certain inequalities, which provide necessary and sufficient conditions for the realizability of that degree sequence. This theorem is crucial for understanding how the degrees of vertices relate to the structure of graphs, linking closely with properties like bipartiteness and regularity.
Graph Coloring: Graph coloring is the assignment of labels, often referred to as 'colors,' to the vertices of a graph in such a way that no two adjacent vertices share the same color. This concept is vital in various applications such as scheduling, resource allocation, and map coloring. The minimum number of colors required to achieve this is called the chromatic number of the graph and plays a crucial role in the study of graph properties and structures.
Graphical Sequences: A graphical sequence is a list of non-negative integers that can represent the degree sequence of a simple graph, meaning it can be realized as a graph with those degrees for each vertex. This concept is crucial when discussing different types of graphs, such as bipartite, complete, and regular graphs, as it helps determine the existence of such graphs based on their degree sequences. By analyzing whether a sequence is graphical, one can understand the structural properties and characteristics of the corresponding graph.
Havel-Hakimi Algorithm: The Havel-Hakimi Algorithm is a method used to determine if a given degree sequence can represent a simple graph. It works by repeatedly reducing the sequence and checking for its feasibility, connecting closely to the concepts of degree sequences and their properties in graph theory. This algorithm not only helps in validating degree sequences but also serves as a foundation for understanding special types of graphs, such as bipartite, complete, and regular graphs.
K-regular graph: A k-regular graph is a type of graph where each vertex has exactly k edges connecting it to other vertices. This uniformity means that every vertex is degree k, which helps establish certain structural properties within the graph. These graphs can be useful for modeling networks and are often studied in relation to special types such as bipartite and complete graphs, as they help illustrate various connectivity and balance features.
Matching problems: Matching problems refer to combinatorial problems where the objective is to pair elements from one set with elements from another set under certain constraints. These problems often arise in various contexts, such as assigning tasks to workers or pairing students with advisors, and can be solved using techniques from graph theory or combinatorial optimization. In many cases, these problems can be represented using bipartite graphs, where the two sets of elements are the vertices, and edges represent valid matches between them.
Odd-length cycles: Odd-length cycles are closed paths in a graph where the number of edges is odd, meaning they cannot be divided evenly into pairs. These cycles have unique properties that affect the structure and characteristics of a graph, especially in relation to its coloring and bipartiteness. An important aspect of odd-length cycles is that they indicate that a graph cannot be bipartite, as bipartite graphs only contain even-length cycles.
Petersen Graph: The Petersen graph is a well-known, highly symmetrical graph that has 10 vertices and 15 edges. It is often used in combinatorics due to its interesting properties, such as being 3-regular and non-bipartite. The Petersen graph serves as an important example for various graph theoretical concepts, illustrating the characteristics of special types of graphs, including their connectivity and chromatic number.
Regular Graph: A regular graph is a type of graph where every vertex has the same degree, meaning that each vertex is connected to the same number of edges. This uniformity in vertex connections leads to interesting properties and structures in graph theory, and it allows for easy application of certain theorems like the Handshaking Lemma, which relates to the total degree of the graph. Regular graphs are also a specific category within special types of graphs, such as complete and bipartite graphs.
Simple undirected graph: A simple undirected graph is a type of graph that consists of a set of vertices connected by edges, where each edge connects two distinct vertices and no two edges connect the same pair of vertices. This means there are no loops (edges that connect a vertex to itself) and no multiple edges between any two vertices. In this context, it serves as a foundation for understanding special types of graphs, highlighting the relationships and properties that can be derived from such basic structures.
Strongly regular graphs: Strongly regular graphs are a special class of graphs characterized by their highly structured nature, defined by parameters that dictate how vertices are connected. These graphs have a fixed number of vertices, and each vertex has the same degree, which contributes to their regularity. The uniqueness of strongly regular graphs lies in their specific conditions for adjacency between vertices, making them an interesting study in combinatorial design and graph theory.
Threshold Graphs: Threshold graphs are a special class of graphs characterized by the property that their vertex set can be partitioned into two subsets: one with a small number of vertices and the other with a large number, where each vertex in the larger subset is connected to all vertices in the smaller subset. This structure connects closely with bipartite graphs, complete graphs, and regular graphs, as threshold graphs can be seen as a simplified form that preserves certain relationships between these different types of graphs.
Two-coloring algorithm: The two-coloring algorithm is a method used to determine whether a graph can be colored using only two colors such that no two adjacent vertices share the same color. This algorithm is particularly useful for identifying bipartite graphs, where the vertex set can be divided into two disjoint sets with edges only connecting vertices from different sets. Successfully applying this algorithm reveals important properties about the structure of the graph and its potential applications in various fields.
Vertex degree: The vertex degree of a graph is the number of edges incident to a vertex, representing how many connections that vertex has with other vertices in the graph. This concept is crucial for understanding the structure and properties of different types of graphs, such as bipartite graphs, complete graphs, and regular graphs. Each of these graph types has distinct characteristics related to vertex degrees, which can help reveal patterns in connectivity and can influence algorithms used for network analysis.