Isomorphisms and Automorphisms
Two graphs G and H are considered identical, denoted as G = H, if they have the same set of vertices V(G) = V(H), the same set of edges E(G) = E(H), and their corresponding properties ψ G = ψ H are equal While identical graphs can be represented by the same diagrams, it is also possible for non-identical graphs to appear similar in their diagrams For instance, graphs G and H may have identical structures, as shown in Figure 1.6, where their diagrams look the same but differ only in the labeling of vertices and edges Such graphs are termed isomorphic despite not being identical.
In general, two graphsGand H are isomorphic, written G∼=H, if there are bijections θ:V(G)→V(H) and φ:E(G)→E(H) such thatψ G (e) =uv if and only if ψ H (φ(e)) = θ(u)θ(v); such a pair of mappings is called an isomorphism betweenGandH. a b d c e 1 e 2 e 3 e 4 e 5 e 6
To demonstrate that two graphs are isomorphic, it is essential to establish an isomorphism between them The mappings (θ, φ) defined as θ: a b c d w z y x and φ: e 1 e 2 e 3 e 4 e 5 e 6 f 3 f 4 f 1 f 6 f 5 f 2 serve as an isomorphism between the graphs G and H, as illustrated in Figure 1.6.
In simple graphs, isomorphism can be defined succinctly: if (θ, φ) represents an isomorphism between simple graphs G and H, the mapping φ is uniquely determined by θ Specifically, for any edge e = uv in G, φ(e) is given by φ(e) = θ(u)θ(v) Therefore, an isomorphism between two simple graphs G and H can be characterized as a bijection θ: V(G) → V(H) that preserves adjacency, meaning vertices u and v are adjacent in G if and only if their images θ(u) and θ(v) are adjacent in H.
Consider, for example, the graphsGandH in Figure 1.7.
The mapping θ:1 2 3 4 5 6 b d f c e a is an isomorphism betweenGandH, as is
Isomorphic graphs must have the same number of vertices and edges; however, having equal vertex and edge counts does not ensure isomorphism For example, two graphs with eight vertices and twelve edges, as illustrated in Figure 1.8, are not isomorphic This is evidenced by the presence of four mutually nonadjacent vertices (v1, v3, v6, and v8) in graph G If an isomorphism θ existed between graphs G and H, the corresponding vertices θ(v1), θ(v3), θ(v6), and θ(v8) in graph H would also need to be mutually nonadjacent However, it can be verified that no four vertices in graph H are mutually nonadjacent, leading to the conclusion that graphs G and H are not isomorphic.
If two graphs are isomorphic, they are essentially the same in structure, differing only in the naming of their vertices and edges This structural similarity leads us to often omit labels when drawing graphs Formally, an unlabelled graph can be defined as a representative of an equivalence class of isomorphic graphs Labels are primarily assigned to vertices and edges for reference purposes, such as in proofs.
There is a unique complete graph, denoted K n, for any number of vertices n, up to isomorphism Likewise, for two positive integers m and n, there exists a unique complete bipartite graph, represented as K m,n, with part sizes m and n, also up to isomorphism.
In the notation presented, the graphs illustrated in Figure 1.2 are identified as K5, K3,3, and K1,5 Additionally, for any positive integer n, there exists a unique path graph, denoted as Pn, and a unique cycle graph, denoted as Cn, with n vertices The graphs shown in Figure 1.3 represent P4 and C5, respectively.
Determining whether two graphs with n vertices are isomorphic is theoretically possible, but the process can be impractical due to the sheer number of bijections (n!) that must be checked For simple graphs G and H, one could examine each of the n! mappings between their vertex sets, but this brute-force method becomes unmanageable even for moderately sized graphs, such as those with n=100 While the number of bijections may decrease for non-regular graphs, where isomorphisms must map vertices of the same degree, this reduction is often insufficient Currently, no efficient general procedure exists for isomorphism testing; however, Luks (1982) introduced an effective algorithm utilizing group-theoretic methods specifically for cubic graphs and graphs with bounded maximum degree.
There is another important matter related to algorithmic questions such as graph isomorphism Suppose that two simple graphs G and H are isomorphic.
Finding an isomorphism between two structures can be challenging; however, once an isomorphism θ is established, verifying its validity becomes straightforward To confirm that θ is an isomorphism, one simply needs to check its properties for each of the n elements involved.
Verifying whether two graphs are isomorphic can be a challenging task, especially when they are not In the case of isomorphic graphs, a bijection between their vertex sets can be established, where two pairs of vertices are connected if and only if their corresponding vertices in the other graph are also connected However, when the graphs are not isomorphic, determining this fact can be just as difficult as determining isomorphism, requiring a thorough examination of all possible bijections or identification of distinct structural properties between the two graphs.
An automorphism of a graph refers to an isomorphism that maps the graph onto itself In the context of a simple graph, this means that an automorphism is a permutation of the vertex set that maintains the adjacency of vertices; specifically, if there is an edge between vertices u and v, the permuted vertices α(u) and α(v) will also have an edge between them.
The automorphisms of a graph highlight its symmetries, indicating that if there is an automorphism α mapping one vertex to another, those vertices are considered similar In graphs where all vertices are similar, such as complete graphs, this property is particularly evident.
The complete bipartite graph K_n, the n-cube Q_n, and the graph K_n,n are classified as vertex-transitive graphs In contrast, asymmetric graphs have no two vertices that are similar and possess only the identity permutation as their automorphism.
Particular drawings of a graph may often be used to display its symmetries.
The Petersen graph, illustrated in three different drawings in Figure 1.9, possesses numerous unique properties It is an exercise (1.2.5) to confirm that all three representations depict the same graph Notably, the first drawing highlights the similarity among the five vertices of the outer pentagon.
The Petersen graph displays vertex-transitivity, as all ten of its vertices are similar This is evident through the rotational symmetry of the inner pentagon's five vertices and the reflective or rotational symmetry of the outer hexagon's six vertices.
Fig 1.9.Three drawings of the Petersen graph
Graphs Arising from Other Structures
Engaging graphs can be created from geometric and algebraic elements, with many constructions being relatively simple However, some may require a deeper understanding and insight to achieve effectively.
A polyhedral graph represents the 1-skeleton of a polyhedron, consisting of vertices and edges that correspond directly to those of the polyhedron, maintaining the same incidence relations Notably, the five Platonic solids—the tetrahedron, cube, octahedron, dodecahedron, and icosahedron—generate their respective Platonic graphs For these classical polyhedra, the graphs are commonly referred to by the same names as the polyhedra they originate from.
Fig 1.14 The five platonic graphs: (a) the tetrahedron, (b) the octahedron, (c) the cube, (d) the dodecahedron, (e) the icosahedron
A set system is defined as an ordered pair (V,F), where V represents a collection of elements and F is a family of subsets of V When F comprises pairs of elements from V, the set system (V,F) forms a loopless graph, making set systems a generalization of graphs, commonly known as hypergraphs In this context, the elements of V are termed the vertices, while the elements of F are referred to as edges or hyperedges A hypergraph is considered k-uniform if each edge is a k-set, meaning it contains k elements Set systems can generate graphs primarily through two methods: incidence graphs and intersection graphs.
Many interesting examples of hypergraphs are provided by geometric config- urations A geometric configuration (P,L) consists of a finite set P of elements
In geometric configurations, a set of points is represented as a graph, while a finite collection of subsets known as lines ensures that no two points share more than one line Notable examples of such configurations include the Fano plane and the Desargues configuration.
These two configurations are shown in Figure 1.15 In both cases, each line consists of three points These configurations thus give rise to 3-uniform hypergraphs; the
Fano hypergraphhas seven vertices and seven edges, theDesargues hypergraphten vertices and ten edges.
Fig 1.15 (a) The Fano plane, and (b) the Desargues configuration
The Fano plane represents the simplest form of projective planes, a significant category of geometric configurations It is closely related to the Desargues configuration, which is derived from a fundamental theorem in projective geometry For more intriguing geometric configurations, refer to works by Coxeter (1950) and Godsil and Royle (2001).
A natural graph linked to a set system H = (V, F) is known as the bipartite graph G[V, F], where vertices v ∈ V and sets F ∈ F are adjacent if v belongs to F This bipartite graph G is referred to as the incidence graph of the set system H, while its bipartite adjacency matrix is termed the incidence matrix of H These representations provide alternative ways to depict a set system In the realm of geometric configurations, incidence graphs often lead to intriguing bipartite graphs, and in this context, the incidence graph may be referred to differently.
Levi graph of the configuration The incidence graph of the Fano plane is shown in Figure 1.16 This graph is known as theHeawood graph.
Each set system (V, F) can be linked to its corresponding intersection graph, where the vertex set is defined by F In this graph, two sets within F are considered adjacent if they share a nonempty intersection For example, when V represents the vertex set of a simple graph G and F is equal to E, this relationship becomes particularly evident.
The incidence graph of the Fano plane, represented by the Heawood graph, illustrates the relationship between the edges of graph G In this context, the intersection graph of the vertex set V and edge set F features edges of G as its vertices, with adjacency defined by shared endpoints This graph is historically referred to as the line graph of G and is denoted as L(G) Additionally, Figure 1.17 showcases both a graph and its corresponding line graph.
Fig 1.17.A graph and its line graph
The intersection graph of the Desargues configuration is isomorphic to the line graph of K5, which is also isomorphic to the complement of the Petersen graph In contrast, the intersection graph of the Fano plane is isomorphic to K7, as every pair of its seven lines shares a common point Furthermore, the definition of the line graph L(G) can be applied to all loopless graphs.
Gas being the graph with vertex setE in which two vertices are joined by just as many edges as their number of common ends in G.
An interval graph is defined as the intersection graph of a set of closed intervals on the real line, denoted as (V, F) Practical examples of interval graphs are discussed in Berge's 1973 book, which also features a detective story in which the resolution depends on the principles of interval graph theory, as noted in Berge's 1995 work.
Graphs are integral to a diverse range of structures, offering valuable insights into their origins Many of these graphs are not only intriguing on their own but also enhance our understanding of the underlying frameworks.
1.3.1 a) Show that the graph in Figure 1.18 is isomorphic to the Heawood graph (Fig- ure 1.16).
Fig 1.18.Another drawing of the Heawood graph b) Deduce that the Heawood graph is vertex-transitive.
1.3.2Show that the following three graphs are isomorphic: the intersection graph of the Desargues configuration, the line graph ofK 5, the complement of the Petersen graph.
1.3.3Show that the line graph ofK 3,3 is self-complementary.
1.3.4Show that neither of the graphs displayed in Figure 1.19 is a line graph.
In a hypergraph H represented by the pair (V, F), the degree of a vertex v, denoted as d(v), refers to the number of edges incident to that vertex The degree sequence of H is defined as a vector d = (d(v) : v ∈ V) The incidence matrix M of H captures the relationships between vertices and edges It can be demonstrated that the sum of the columns of the incidence matrix M equals the degree sequence d, illustrating the connection between the matrix representation and the vertex degrees in the hypergraph.
1.3.6LetH := (V,F) be a hypergraph Forv∈V, letF v denote the set of edges ofH incident tov ThedualofH is the hypergraphH ∗ whose vertex set isF and whose edges are the setsF v ,v∈V.
The relationship between the incidence graphs of hypergraphs H and H* is explored, highlighting their connection It is demonstrated that the dual of H* is isomorphic to H, establishing a fundamental property of hypergraphs Additionally, the concept of self-duality in hypergraphs is defined, with specific focus on the Fano and Desargues hypergraphs, which are proven to be self-dual.
A family of sets is said to possess the Helly Property if every pairwise intersecting subfamily shares at least one common element To illustrate this, we can demonstrate that the family of closed intervals on the real line satisfies the Helly Property, as any collection of closed intervals that intersect in pairs will necessarily have a point that lies within all of them.
(E Helly) b) Deduce that the graph in Figure 1.20 is not an interval graph.
Fig 1.20.A graph that is not an interval graph
In graph theory, let \( m \) and \( n \) be positive integers with \( n > 2m \) The Kneser graph \( KG_{m,n} \) is defined with vertices representing the \( m \)-subsets of an \( n \)-set \( S \), where two subsets are adjacent if they have no elements in common It can be demonstrated that \( KG_{1,n} \) is isomorphic to the complete graph \( K_n \) for \( n \geq 3 \) Additionally, for \( n \geq 5 \), the Kneser graph \( KG_{2,n} \) is isomorphic to the complement of the line graph \( L(K_n) \).
Constructing Graphs from Other Graphs
We have already seen a couple of ways in which we may associate with each graph another graph: the complement (in the case of simple graphs) and the line graph.
When considering two simple graphs, G and H, rather than a single graph, we can define a new graph in multiple ways For clarity, we denote G and H as simple graphs, where each edge consists of an unordered pair of vertices The concepts discussed can be easily adapted to more complex scenarios.
Two graphs are considered disjoint if they share no vertices, while they are edge-disjoint if they share no edges The fundamental methods for combining graphs include union and intersection The union of two simple graphs, G and H, is represented as G∪H, incorporating the vertex set V(G)∪V(H) and the edge set E(G)∪E(H) When G and H are disjoint, their union is termed a disjoint union, typically denoted as G+H These operations are both associative and commutative, allowing extension to multiple graphs A graph is classified as disconnected if it can be expressed as a disjoint union of two non-null graphs.
A graph G can be uniquely represented as a disjoint union of connected graphs, known as its connected components, with the number of these components denoted as c(G) Notably, the null graph is unique in that it has no components Similarly, the intersection of two graphs G and H, denoted as G∩H, is defined in an analogous manner.
Gand H are disjoint, their intersection is the null graph.) Figure 1.22 illustrates these concepts The graph G∪H shown in Figure 1.22 has just one component, whereas the graphG∩H has two components.
Fig 1.22.The union and intersection of two graphs
Several methods exist to create a new graph from two given graphs, where the vertex set of the new graph is the Cartesian product of the original vertex sets These constructions are commonly known as "products." Here, we will outline one of these methods.
The Cartesian product of simple graphs G and H, denoted as G H, is defined by its vertex set V(G) × V(H) and an edge set that includes all pairs (u₁, v₁)(u₂, v₂) where either u₁u₂ is an edge in G with v₁ = v₂, or v₁v₂ is an edge in H with u₁ = u₂ Consequently, for each edge u₁u₂ in G and each edge v₁v₂ in H, four edges are formed in GH: (u₁, v₁)(u₂, v₁), (u₁, v₂)(u₂, v₂), (u₁, v₁)(u₁, v₂), and (u₂, v₁)(u₂, v₂) More broadly, the Cartesian product Pₘ Pₙ of two paths results in an (m×n)-grid.
Fig 1.23.(a) The cartesian productK 2 K 2, and (b) the (5×4)-grid
For any integer n ≥ 3, the Cartesian product C_n × K_2 forms a polyhedral graph known as an n-prism Specifically, the 3-prism, 4-prism, and 5-prism are referred to as the triangular prism, cube, and pentagonal prism, respectively The Cartesian product is considered one of the fundamental types of graph products, with several other variations that emerge in different contexts, which will be explored in subsequent chapters.
Fig 1.24.The triangular and pentagonal prisms
1.4.1Show that every graph may be expressed uniquely (up to order) as a disjoint union of connected graphs.
1.4.2Show that the rank overGF(2) of the incidence matrix of a graphGisn−c.
1.4.3Show that the cartesian product is both associative and commutative.
1.4.4Find an embedding of the cartesian productC m C n on the torus.
The Cartesian product of two vertex-transitive graphs is itself vertex-transitive, demonstrating that the symmetry properties of these graphs are preserved in their product However, it is important to note that the Cartesian product of two edge-transitive graphs does not guarantee edge-transitivity, highlighting a distinction in the behavior of these graph properties.
Let G be a self-complementary graph, and consider a path P of length three that is disjoint from G By constructing a new graph H from the union of G and P, where the first and third vertices of P are connected to each vertex of G, it can be demonstrated that H remains self-complementary Furthermore, by referencing Exercise 1.2.16, it can be concluded that a self-complementary graph with n vertices exists if and only if n is congruent to 0 or 1 modulo 4.
Directed Graphs
While graph theory is useful for various problems, it may not always be sufficient In traffic flow scenarios, understanding one-way roads and their permitted directions is crucial Therefore, a simple graph is inadequate; instead, a directed graph, where each link has a specific orientation, is essential for accurately representing the network.
Formally, adirected graphDis an ordered pair (V(D), A(D)) consisting of a set
In a directed graph D, the set of vertices is represented as V(D) and the set of arcs as A(D), which is disjoint from V(D) An incidence function ψ_D links each arc to an ordered pair of vertices, indicating that if an arc a has ψ_D(a) = (u, v), then a connects u to v, with u dominating v Here, u is referred to as the tail and v as the head of the arc In discussions where the orientation of an arc is not significant, it is labeled as an edge The total number of arcs in D is denoted by a(D) Vertices that dominate a vertex v are termed its in-neighbours, while those dominated by v are its out-neighbours, represented by N_D^-(v) and N_D^+(v), respectively.
For convenience, we abbreviate the term ‘directed graph’ todigraph Astrict digraph is one with no loops or parallel arcs (arcs with the same head and the same tail).
In any digraph D, we can create an associated graph G by converting each arc into an edge with identical endpoints, resulting in the underlying graph G(D) Conversely, any graph G can be transformed into a digraph by turning each edge into two oppositely directed arcs, known as the associated digraph D(G) Additionally, a digraph can be derived from a graph G by replacing each edge with one of the two possible arcs at the same endpoints, referred to as an orientation of G.
An orientation of a simple graph, known as an oriented graph, specifies a particular direction for its edges A notable example of this is the orientation of a complete graph, which is termed a tournament This type of oriented graph effectively represents the outcomes of a round-robin tournament, where each team competes against every other team without any ties.
Digraphs, similar to graphs, feature a straightforward visual representation They are depicted by a diagram of their underlying graph, with arrows on the edges indicating the direction of each arc Figure 1.25 illustrates the four unlabelled tournaments involving four vertices.
Fig 1.25 The four unlabelled tournaments on four vertices
Every concept that is valid for graphs automatically applies to digraphs too. For example, thedegree of a vertexv in a digraph D is simply the degree of v in
A digraph is considered connected if its underlying graph is also connected In digraphs, the orientation of arcs is crucial, as demonstrated by the indegree \(d^{-}(v)\) of a vertex \(v\), which counts the arcs directed towards \(v\), and the outdegree \(d^{+}(v)\), which counts the arcs directed away from \(v\) The minimum indegree and outdegree of a digraph \(D\) are represented as \(\delta^{-}(D)\) and \(\delta^{+}(D)\), while the maximum values are similarly defined.
In certain situations, we use the same notation as that of graphs, substituting G with D Consequently, the degree of a vertex v in D is represented as dD(v) This consistent notation is documented only once in the glossaries, specifically for graphs.
The index focuses solely on definitions of digraphs that differ significantly from those of graphs, omitting terms like 'connected digraph' in favor of 'connected graph.' A digraph is termed k-regular if every vertex has equal indegree and outdegree of k Vertices with an indegree of zero are identified as sources, while those with an outdegree of zero are called sinks Directed paths and directed cycles represent orientations where each vertex leads to its successor Additionally, the concept of connectedness in digraphs considers the direction of edges, which will be further explored in Chapter 2.
Figure 1.26 illustrates two unique digraphs: a 2-regular digraph and a 3-regular digraph, following the convention of representing two oppositely oriented arcs as a single edge Both digraphs can be derived from the Fano plane, as detailed in Exercise 1.5.9, and they exhibit additional intriguing properties that will be explored in Chapter 2.
Fig 1.26 (a) the Koh–Tindell digraph, and (b) a directed analogue of the Petersen graph
Interesting digraphs can be derived from various mathematical structures, such as groups, offering a natural directed analogue of a Cayley graph A Cayley digraph, denoted as CD(Γ, S), can be formed from a group Γ and a subset S, excluding the identity element, where vertex x dominates vertex y if xy - 1 is an element of S A specific type of Cayley digraph is the directed circulant, represented as CD(Zn, S), where Zn is the group of integers modulo n.
Every digraph D can be associated with a converse digraph, denoted as ←D−, which is created by reversing each arc of D This converse digraph is essentially the directional dual of the original digraph, as taking the converse of a digraph twice returns the original structure This relationship highlights a straightforward yet valuable principle in the study of digraphs.
Any statement about a digraph has an accompanying ‘dual’ statement, obtained by applying the statement to the converse of the digraph and reinterpreting it in terms of the original digraph.
In a directed graph, the total sum of the indegrees of all vertices is equal to the total number of arcs By applying the Principle of Directional Duality, we can also conclude that the sum of the outdegrees is equal to the total number of arcs.
Assigning appropriate orientations to the edges of a graph not only serves practical purposes but also facilitates the exploration of the graph's properties, which will be discussed in Chapter 6.
1.5.1How many orientations are there of a labelled graphG?
1.5.2LetD be a digraph. a) Show that v∈V d − (v) =m. b) Using the Principle of Directional Duality, deduce that v∈V d + (v) =m.
Two digraphs D and D are considered isomorphic, denoted as D ∼= D, if there exist bijections θ: V(D) → V(D) and φ: A(D) → A(D) such that the mapping ψ_D(a) = (u, v) holds if and only if ψ_D(φ(a)) = (θ(u), θ(v)) This relationship defines an isomorphism between the digraphs In this context, it can be demonstrated that the four tournaments depicted in Figure 1.25 are pairwise nonisomorphic, and these represent the only tournaments on four vertices, up to isomorphism Additionally, an inquiry into the number of tournaments on five vertices, considering isomorphism, is posed.
Vertex-transitivity and arc-transitivity are essential concepts in the study of directed graphs (digraphs) A digraph is considered vertex-transitive if, for any two vertices, there exists a directed path that allows one to reach the other, indicating uniformity in vertex connections On the other hand, a digraph is arc-transitive if, for any two directed edges, there is a directed path that maintains the same directionality It is important to note that every vertex-transitive digraph is diregular, meaning all vertices have the same in-degree and out-degree An example of a vertex-transitive digraph is the Koh–Tindell digraph, which demonstrates vertex-transitivity but lacks arc-transitivity, highlighting the distinction between these two properties in digraphs.
1.5.5A digraph isself-converseif it is isomorphic to its converse Show that both digraphs in Figure 1.26 are self-converse.
Let D be a digraph with vertex set V and arc set A The incidence matrix of
D (with respect to given orderings of its vertices and arcs) is the n×mmatrix
1 if arcais a link and vertexv is the tail ofa
−1 if arcais a link and vertexv is the head ofa
0 otherwise LetM be the incidence matrix of a connected digraph D Show that the rank of
Infinite Graphs
This book focuses on finite graphs, but it also acknowledges the existence of infinite graphs, which are defined on infinite sets of vertices and/or edges An infinite graph is classified as countable if both its vertex and edge sets are countable Notable examples of countable graphs include the square lattice, triangular lattice, and hexagonal lattice, as illustrated in Figure 1.27.
Fig 1.27.The square, triangular and hexagonal lattices
Many principles applicable to finite graphs can be adapted for infinite graphs with minor adjustments The degree of a vertex remains similar to that in finite graphs, with 'number' replaced by 'cardinality.' Infinite paths are categorized into two types: one-way infinite paths, which have an initial vertex but no terminal vertex, and two-way infinite paths, which lack both For example, the square lattice is formed by the Cartesian product of two two-way infinite paths However, some concepts in finite graphs, like cycles, do not have a direct infinite counterpart, although a two-way infinite path may occasionally be viewed as an infinite cycle under specific conditions.
This book primarily concentrates on finite graphs, but it also features remarks and exercises related to infinite graphs to highlight the distinctions between the two For readers seeking more information on infinite graphs, the survey article by Thomassen (1983a) and Diestel's book (2005), which contains a chapter dedicated to the topic, are recommended resources.
An infinite graph islocally finiteif every vertex is of finite degree Give an example of a locally finite graph in which no two vertices have the same degree.
For every positive integer d, there exists a simple infinite planar graph characterized by a minimum degree of d In Chapter 10, we will explore the fact that every simple finite planar graph contains at least one vertex with a degree of five or less.
1.6.3Give an example of a self-complementary infinite graph.
The unit distance graph on a subset V of R² consists of vertices where two points (x₁, y₁) and (x₂, y₂) are adjacent if their Euclidean distance is exactly 1, represented mathematically as (x₁ - x₂)² + (y₁ - y₂)² = 1 When V is the set of rational numbers Q², it forms the rational unit distance graph, while V being the set of real numbers R² creates the real unit distance graph; both are infinite graphs For a finite subset V of the 2-dimensional integer lattice and an odd positive integer d, the graph G is defined with vertices in V, where adjacency is based on an Euclidean distance of d It can be shown that G is bipartite, leading to the conclusion that the rational unit distance graph is also bipartite Conversely, it can be demonstrated that the real unit distance graph does not exhibit bipartiteness.
Related Reading
The history of graph theory up to 1936 is compellingly detailed in Biggs et al (1986), featuring key excerpts from significant papers The publication of the first book on graph theory by König in 1936 marked a pivotal moment, fostering a robust community of graph theorists in Hungary, including notable figures like P Erdős.
T Gallai Also in the thirties, H Whitney published a series of influential articles (see Whitney (1992)).
Graph theory, like all areas of mathematics, is most effectively learned through practice For those seeking engaging problems and proof techniques, Lovász's book, "Combinatorial Problems and Exercises" (1993), is highly recommended Additionally, the classic guide "How to Solve" offers valuable insights into problem-solving in mathematics.
It by P´olya (2004) The delightful Proofs from the Book by Aigner and Ziegler
(2004) is a compilation of beautiful proofs in mathematics, many of which treat combinatorial questions.
Edge and Vertex Deletion 40 Maximality and Minimality 41 Acyclic Graphs and Digraphs 42 Proof Technique: The Pigeonhole Principle 43
Spanning Subgraphs 46 Proof Technique: Induction 48 Proof Technique: Contradiction 49 Induced Subgraphs 49 Weighted Graphs and Subgraphs 50
Vertex Identification and Edge Contraction 55 Vertex Splitting and Edge Subdivision 55
Decompositions 56 Proof Technique: Linear Independence 57 Coverings 58
Edge Cuts 59 Bonds 62 Cuts in Directed Graphs 62
The Cycle and Bond Spaces 65
The Reconstruction Conjecture 67 The Edge Reconstruction Conjecture 68 Proof Technique: M¨obius Inversion 68
Path and Cycle Decompositions 76Legitimate Decks 76Ultrahomogeneous Graphs 77
Given a graph G, there are two natural ways of deriving smaller graphs fromG.
If e is an edge ofG, we may obtain a graph on m−1 edges by deleting e from
In graph theory, removing edges or vertices from a graph can lead to new graph configurations When we delete an edge from a graph \( G \), the resulting graph is represented as \( G \setminus e \), retaining all vertices and remaining edges Conversely, deleting a vertex \( v \) along with its incident edges results in a graph denoted as \( G \setminus v \), which contains \( n-1 \) vertices These concepts of edge deletion and vertex deletion are visually demonstrated in Figure 2.1.
Fig 2.1.Edge-deleted and vertex-deleted subgraphs of the Petersen graph
In graph theory, the edge-deleted subgraph G\e and the vertex-deleted subgraph G−v are specific examples of subgraphs derived from a graph G A graph F is classified as a subgraph of G if its vertex set V(F) is a subset of V(G) and its edge set E(F) is a subset of E(G), with the function ψ F being the restriction of ψ G to E(F) Consequently, we express the relationship between G and F by stating that G contains F, denoted as G⊇F, or that F is contained in G, written as F⊆G.
The graph F of G can be derived through successive edge and vertex deletions, starting with the removal of edges in G that are not part of F, followed by the elimination of vertices in G that do not belong to F It is important to note that the null graph is considered a subgraph of every graph.
In the specific scenario where a graph G is vertex-transitive, all subgraphs formed by deleting a vertex from G are isomorphic In this context, we use the notation G−v to represent any vertex-deleted subgraph Similarly, for an edge-transitive graph G, we denote any edge-deleted subgraph as G\e.
A copy of a graph F in a graph G is a subgraph of G which is isomorphic to F Such a subgraph is also referred to as an F-subgraph of G; for instance, a
K 3-subgraph is a triangle in the graph Anembedding of a graphF in a graphG is an isomorphism betweenF and a subgraph ofG For each copy ofF inG, there are aut(F) embeddings of F in G.
Asupergraphof a graphGis a graphH which containsGas a subgraph, that is, H ⊇ G Note that any graph is both a subgraph and a supergraph of itself.
All other subgraphsF and supergraphsH are referred to asproper; we then write
The above definitions apply also to digraphs, with the obvious modifications.
In graph theory, determining whether a graph contains a specific subgraph or supergraph with desired properties is crucial A significant theorem provides a sufficient condition for a graph to have a cycle Subsequent chapters will explore the criteria for graphs that include long paths, cycles, or complete subgraphs of specified orders While supergraphs with specific properties are less common, they naturally emerge in various applications, as discussed in Chapter 16 and referenced in Exercises 2.2.17 and 2.2.24.
Theorem 2.1 LetGbe a graph in which all vertices have degree at least two Then
If graph G contains a loop, it inherently has a cycle of length one, and if it has parallel edges, it contains a cycle of length two, allowing us to assume that G is simple Let P denote the longest path in G, represented as v0 v1 vk-1 vk Since the degree of vertex vk is at least two, it must have a neighbor, v, different from vk-1 If v is not part of path P, then the extended path v0 v1 vk-1 vk v contradicts the assumption that P is the longest path Consequently, v must be equal to vi for some index i, where 0 ≤ i ≤ k-2, resulting in the cycle vi vi+1 vk vi within G.
The proof of Theorem 2.1 begins by identifying a longest path in the graph, which leads to the discovery of a cycle While this method is mathematically sound, locating the longest path can be challenging However, the proof remains valid if we consider a maximal path instead, which is defined as a path that cannot be extended from either end Finding a maximal path is straightforward; one can start at any vertex and extend the path until no further extension is possible This highlights the significance of maximality and minimality in the study of subgraphs.
In graph theory, a family of subgraphs F of a graph G can have maximal and minimal members, where a maximal member is one that is not properly contained by any other member, and a minimal member is one that is not properly containing any other member For example, when F includes all paths in G, a maximal member is termed a maximal path Similarly, when F represents all connected subgraphs, the maximal members are referred to as components Notably, odd cycles serve as minimal nonbipartite subgraphs since they are not bipartite themselves, while all their proper subgraphs are bipartite In fact, odd cycles are the only minimal nonbipartite subgraphs in a graph.
Fig 2.2.(a) A maximal path, (b) a minimal nonbipartite subgraph, and (c) a maximal bipartite subgraph
Maximality and minimality should not be confused with maximum and minimum cardinality in graph theory Every cycle in a graph is considered a maximal cycle, as no cycle can be contained within another, making it also a minimal cycle Conversely, a maximum cycle refers to the longest cycle in a graph, while a minimum cycle denotes the shortest one In a graph \( G \) with at least one cycle, the length of the longest cycle is termed its circumference, and the length of the shortest cycle is referred to as its girth.
An acyclic graph is defined as one that does not contain any cycles, which implies that it must have at least one vertex with a degree of less than two In fact, every nontrivial acyclic graph contains at least two vertices with a degree of less than two Similarly, a directed graph (digraph) is considered acyclic if it lacks directed cycles A noteworthy category of acyclic digraphs is those that correspond to partially ordered sets.
A partially ordered set, or for short poset, is an ordered pair P = (X,≺), where
In a set X with a partial order ≺, which is defined as an irreflexive, antisymmetric, and transitive binary relation, two elements u and v are considered comparable if either u ≺ v or v ≺ u; otherwise, they are deemed incomparable A collection of elements within X that are pairwise comparable forms a chain, while a collection of pairwise incomparable elements is referred to as an antichain.
One can form a digraph D := D(P) from a poset P = (X,≺) by taking
In a directed graph (digraph) X, an arc (u, v) exists if and only if u precedes v, establishing a structure that is both acyclic and transitive Here, transitivity implies that if arcs (u, v) and (v, w) are present, then arc (u, w) must also exist It is important to note that this concept of transitivity in digraphs is unrelated to the group-theoretic ideas of vertex-transitivity and edge-transitivity Additionally, each strict acyclic transitive digraph D corresponds to a partially ordered set (poset).
P on the vertex set of D An acyclic tournament is frequently referred to as a
2.1 Subgraphs and Supergraphs 43 transitive tournament It can be seen that chains in P correspond to transitive subtournaments of D.
Proof Technique: The Pigeonhole Principle
The Pigeonhole Principle states that when distributing \( n + 1 \) letters into \( n \) pigeonholes, at least two letters must occupy the same pigeonhole This principle illustrates a fundamental concept in combinatorics, particularly regarding multisets, which are sets that allow for repeated elements.