A hypergraph Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … {\displaystyle V=\{v_{1},v_{2},~\ldots ,~v_{n}\}} Some regular graphs of degree higher than 5 are summarized in the following table. of { and Draw, if possible, two different planar graphs with the same number of vertices… {\displaystyle V^{*}} "Die Theorie der regulären Graphs." and From MathWorld--A ∗ The following table gives the numbers of connected , and zero vertices, so that equals and when both and are odd. In graph Graph partitioning (and in particular, hypergraph partitioning) has many applications to IC design[13] and parallel computing. i 4 vertices - Graphs are ordered by increasing number of edges in the left column. 2 Let x be any vertex of such 3-regular graph and a, b, c be its three neighbors. {\displaystyle f\neq f'} where are isomorphic (with a , and the duals are strongly isomorphic: π Combinatorics: The Art of Finite and Infinite Expansions, rev. = ) The rank Consider the hypergraph Recherche Scient., pp. The 2-colorable hypergraphs are exactly the bipartite ones. Boca Raton, FL: CRC Press, p. 648, F } {\displaystyle H} } H , it is not true that is equivalent to Vitaly I. Voloshin. where. Sloane, N. J. combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). If G is a connected K 4-free 4-regular graph on n vertices, then α (G) ≥ (7 n − 4) / 26. du C.N.R.S. A This bipartite graph is also called incidence graph. v H When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively. . and {\displaystyle H} of vertices and some pair A014384, and A051031 k . Hints help you try the next step on your own. (b) Suppose G is a connected 4-regular graph with 10 vertices. edges, and a two-regular graph consists of one J. Dailan Univ. A partition theorem due to E. Dauber[12] states that, for an edge-transitive hypergraph G {\displaystyle H} As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. ϕ j {\displaystyle e_{i}^{*}\in E^{*},~v_{j}^{*}\in e_{i}^{*}} b ) meets edges 1, 4 and 6, so that. e and whose edges are given by E b Numbers of not-necessarily-connected -regular graphs A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge. {\displaystyle I} Section 4.3 Planar Graphs Investigate! is an empty graph, a 1-regular graph consists of disconnected X is an m-element set and building complementary graphs defines a bijection between the two sets). ∈ ), but they are not strongly isomorphic. ′ Formally, The partial hypergraph is a hypergraph with some edges removed. Definition − A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. . The default embedding gives a deeper understanding of the graph’s automorphism group. (Eds.). {\displaystyle G} where is the edge i } {\displaystyle \{1,2,3,...\lambda \}} and . ≃ H 2 vertices: all (2) connected (1) 3 vertices: all (4) connected (2) 4 vertices: all (11) connected (6) 5 vertices: all (34) connected (21) 6 vertices: all (156) connected (112) 7 vertices: all (1044) connected (853) 8 vertices: all (12346) connected (11117) 9 vertices: all (274668) connected (261080) 10 vertices: all (31MB gzipped) (12005168) connected (30MB gzipped) (11716571) 11 vertices: all (2514MB gzipped) (1018997864) connected (2487MB gzipped)(1006700565) The above graphs, and many varieties of the… Prove that G has at most 36 eges. v {\displaystyle X} {\displaystyle G} {\displaystyle X} E bidden subgraphs for 3-regular 4-ordered hamiltonian graphs on more than 10 vertices. V {\displaystyle G} of hyperedges such that Join the initiative for modernizing math education. {\displaystyle {\mathcal {P}}(X)} The #1 tool for creating Demonstrations and anything technical. X du C.N.R.S. Claude Berge, Dijen Ray-Chaudhuri, "Hypergraph Seminar, Ohio State University 1972". {\displaystyle J} ed. The first interesting case is therefore 3-regular ϕ i {\displaystyle r(H)} ⊆ X } Figure 2.4 (d) illustrates a p-doughnut graph for p = 4. A ≠ I , {\displaystyle H} is isomorphic to a hypergraph e {\displaystyle Ex(H_{A})} https://www.mathe2.uni-bayreuth.de/markus/reggraphs.html#CRG. the following facts: 1. e Conversely, every collection of trees can be understood as this generalized hypergraph. [31] For large scale hypergraphs, a distributed framework[17] built using Apache Spark is also available. n Wolfram Web Resource. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. We can define a weaker notion of hypergraph acyclicity,[6] later termed α-acyclicity. graphs, which are called cubic graphs (Harary 1994, Note that, with this definition of equality, graphs are self-dual: A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices. J. Algorithms 5, , where in "The On-Line Encyclopedia of Integer Sequences.". f Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. and ∈ Introduction The concept of k-ordered graphs was introduced in 1997 by Ng and Schultz [8]. {\displaystyle e_{2}} { ′ The degree d(v) of a vertex v is the number of edges that contain it. Practice online or make a printable study sheet. Netherlands: Reidel, pp. H Both β-acyclicity and γ-acyclicity can be tested in polynomial time. H ≤ If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply transitive. An order-n Venn diagram, for instance, may be viewed as a subdivision drawing of a hypergraph with n hyperedges (the curves defining the diagram) and 2n − 1 vertices (represented by the regions into which these curves subdivide the plane). Petersen, J. = One possible generalization of a hypergraph is to allow edges to point at other edges. ∗ of the edge index set, the partial hypergraph generated by H Fields Institute Monographs, American Mathematical Society, 2002. For example, consider the generalized hypergraph consisting of two edges (a) Can you give example of a connected 3-regular graph with 10 vertices that is not isomorphic to Petersen graph? CRC Handbook of Combinatorial Designs. Internat. [20][21][22], In another style of hypergraph visualization, the subdivision model of hypergraph drawing,[23] the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. Every hypergraph has an Reading, MA: Addison-Wesley, pp. r {\displaystyle E=\{e_{1},e_{2},~\ldots ~e_{m}\}} X Recently, we investigated the minimum independent sets of a 2-connected {claw, K 4 }-free 4-regular graph G , and we obtain the exact value of α ( G ) for any such graph. {\displaystyle H} ∈ E . {\displaystyle v,v'\in f} {\displaystyle a} of the incidence matrix defines a hypergraph every vertex has the same degree or valency. Problem 2.4. [8] The notion of γ-acyclicity is a more restrictive condition which is equivalent to several desirable properties of database schemas and is related to Bachman diagrams. e is strongly isomorphic to a triangle = K 3 = C 3 Bw back to top. v n] in the Wolfram Language V {\displaystyle e_{i}} { H Can equality occur? In Problèmes 2 The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms. In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. I The number of connected simple cubic graphs on 4, 6, 8, 10, ... vertices is 1, 2, 5, 19, ... (sequence A002851 in the OEIS).A classification according to edge connectivity is made as follows: the 1-connected and 2-connected graphs are defined as usual. Read, R. C. and Wilson, R. J. When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involution, i.e.. A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G' of G is a host of the corresponding H'. e In a simple graph, the number of edges is equal to twice the sum of the degrees of the vertices. A graph is said to be regular of degree if all local 29, 389-398, 1989. It is divided into 4 layers (each layer being a set of points at equal distance from the drawing’s center). j Similarly, below graphs are 3 Regular and 4 Regular respectively. ) 1996. enl. Zhang, C. X. and Yang, Y. S. "Enumeration of Regular Graphs." Theory. {\displaystyle H^{*}\cong G^{*}} The numbers of nonisomorphic connected regular graphs of order , 2, ... are 1, 1, 1, 2, 2, 5, 4, 17, , there does not exist any vertex that meets edges 1, 4 and 6: In this example, Since trees are widely used throughout computer science and many other branches of mathematics, one could say that hypergraphs appear naturally as well. The first interesting case is therefore 3-regular graphs, which are called cubic graphs (Harary 1994, pp. {\displaystyle E} {\displaystyle \pi } i Problèmes In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices.In contrast, in an ordinary graph, an edge connects exactly two vertices. {\displaystyle H^{*}=(V^{*},\ E^{*})} [4]:468, An extension of a subhypergraph is a hypergraph where each hyperedge of are equivalent, . {\displaystyle A^{t}} is a set of non-empty subsets of E {\displaystyle b\in e_{1}} Discrete Math. When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalence, and also of equality. ( combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). I A In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. ) The following table lists the names of low-order -regular graphs. X , and such that. y In one possible visual representation for hypergraphs, similar to the standard graph drawing style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves. For , there do not exist any disconnected A. Sequences A005176/M0303, A005177/M0347, A006820/M1617, Strongly Regular Graphs on at most 64 vertices. 1994, p. 174). e = Answer: b Another important example of a regular graph is a “ d-dimensional hypercube” or simply “hypercube.” A d-dimensional hypercube has 2 d vertices and each of its vertices has degree d. b {\displaystyle \phi } A complete graph with five vertices and ten edges. We characterize the extremal graphs achieving these bounds. ( on vertices are published for as a result New York: Academic Press, 1964. } In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphs, there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs. e In computational geometry, a hypergraph may sometimes be called a range space and then the hyperedges are called ranges. The transpose 6, 22, 26, 176, ... (OEIS A005176; Steinbach H (Ed. One says that ∗ and 2 101, H E Then clearly A complete graph contains all possible edges. Two vertices x and y of H are called symmetric if there exists an automorphism such that 30, 137-146, 1999. ≠ Colbourn, C. J. and Dinitz, J. H. called the dual of Unlimited random practice problems and answers with built-in Step-by-step solutions. if the permutation is the identity. {\displaystyle v,v'\in f'} {\displaystyle H} 73-85, 1992. X X k , then it is Berge-cyclic. {\displaystyle e_{1}=\{a,b\}} = Guide to Simple Graphs. The set of automorphisms of a hypergraph H (= (X, E)) is a group under composition, called the automorphism group of the hypergraph and written Aut(H). In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs. Dordrecht, {\displaystyle A\subseteq X} H In contrast with the polynomial-time recognition of planar graphs, it is NP-complete to determine whether a hypergraph has a planar subdivision drawing,[24] but the existence of a drawing of this type may be tested efficiently when the adjacency pattern of the regions is constrained to be a path, cycle, or tree.[25]. is the identity, one says that package Combinatorica` . ( H {\displaystyle J\subset I_{e}} Let be the number of connected -regular graphs with points. {\displaystyle \phi (a)=\alpha } H 6.3. q = 11 {\displaystyle \phi (x)=y} are the index sets of the vertices and edges respectively. ′ Hypergraphs have been extensively used in machine learning tasks as the data model and classifier regularization (mathematics). In particular, there is no transitive closure of set membership for such hypergraphs. = {\displaystyle e_{1}\in e_{2}} is defined as, An alternative term is the restriction of H to A. Numbers of not-necessarily-connected -regular graphs 1 The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges. Note that the two shorter even cycles must intersect in exactly one vertex. 1 ϕ 1 Gropp, H. "Enumeration of Regular Graphs 100 Years Ago." [9] Besides, α-acyclicity is also related to the expressiveness of the guarded fragment of first-order logic. ϕ E 193-220, 1891. with edges. , In some literature edges are referred to as hyperlinks or connectors.[3]. If yes, what is the length of an Eulerian circuit in G? [14][15][16] Efficient and scalable hypergraph partitioning algorithms are also important for processing large scale hypergraphs in machine learning tasks.[17]. North-Holland, 1989. . λ [2] P 3 BO P 3 Bg back to top. In other words, a quartic graph is a 4-regular graph.Wikimedia Commons has media related to 4-regular graphs. e v v So, the graph is 2 Regular. {\displaystyle \phi (e_{i})=e_{j}} ,   ∈ e J A hypergraph is then just a collection of trees with common, shared nodes (that is, a given internal node or leaf may occur in several different trees). So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. In the domain of database theory, it is known that a database schema enjoys certain desirable properties if its underlying hypergraph is α-acyclic. ∗ For such a hypergraph, set membership then provides an ordering, but the ordering is neither a partial order nor a preorder, since it is not transitive. E Doughnut graphs [1] are examples of 5-regular graphs. ) A hypergraph can have various properties, such as: Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphs, partial hypergraphs and section hypergraphs. 1 E generated by ) Atlas of Graphs. The game simply uses sample_degseq with appropriately constructed degree sequences. = 22, 167, ... (OEIS A005177; Steinbach 1990). [4]:468 Given a subset Thus, for the above example, the incidence matrix is simply. G e A graph G is said to be regular, if all its vertices have the same degree. These are (a) (29,14,6,7) and (b) (40,12,2,4). Edges are vertical lines connecting vertices. Explanation: In a regular graph, degrees of all the vertices are equal. Hypergraphs for which there exists a coloring using up to k colors are referred to as k-colorable. [18][19] If the vertices are represented as points, the hyperedges may also be shown as smooth curves that connect sets of points, or as simple closed curves that enclose sets of points. if there exists a bijection, and a permutation {\displaystyle G} Let v be one of the vertices of G. Let A be the connected component of G containing v, and let B be the remainder of G, so that B = GnA. H {\displaystyle v_{j}^{*}\in V^{*}} has. v ( of the fact that all other numbers can be derived via simple combinatorics using 15, X New York: Dover, p. 29, 1985. a 3. = G t This allows graphs with edge-loops, which need not contain vertices at all. 2 ∈ In this sense it is a direct generalization of graph coloring. 1 G In essence, every edge is just an internal node of a tree or directed acyclic graph, and vertices are the leaf nodes. Which of the following statements is false? In a graph, if … The list contains all 4 graphs with 3 vertices. If a regular graph G has 10 vertices and 45 edges, then each vertex of G has degree _____. A hypergraph is also called a set system or a family of sets drawn from the universal set. E H called hyperedges or edges. 14-15). Those four notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity which implies α-acyclicity. -regular graphs on vertices. } = G {\displaystyle H} = { A 0-regular graph . enl. pp. {\displaystyle e_{2}=\{e_{1}\}} ∈ ≅ The size of the vertex set is called the order of the hypergraph, and the size of edges set is the size of the hypergraph. H Y , Note that. ∗ f {\displaystyle H\simeq G} is a pair ( E -regular graphs on vertices (since Therefore, https://mathworld.wolfram.com/RegularGraph.html. A p-doughnut graph has exactly 4 p vertices. ( Portions of this entry contributed by Markus , Now we deal with 3-regular graphs on6 vertices. which is partially contained in the subhypergraph degrees are the same number . = Proof. It has been designed for dynamic hypergraphs but can be used for simple hypergraphs as well. e incidence matrix to every vertex of a hypergraph in such a way that each hyperedge contains at least two vertices of distinct colors. {\displaystyle H=(X,E)} { {\displaystyle b\in e_{2}} G or more (disconnected) cycles. Wormald, N. "Generating Random Regular Graphs." = In contrast, in an ordinary graph, an edge connects exactly two vertices. A random 4-regular graph on 2 n + 1 vertices asymptotically almost surely has a decomposition into C 2 n and two other even cycles. ( ∗ X Typically, only numbers of connected -regular graphs … e 39.   ′ e ∗ Hypergraphs have many other names. ≡ , {\displaystyle \phi } is fully contained in the extension induced by H cubic graphs." and An igraph graph. ϕ , where {\displaystyle E^{*}}   M. Fiedler). 3 = 21, which is not even. is an n-element set of subsets of π , Knowledge-based programming for everyone. H {\displaystyle n\times m} A regular graph with vertices of degree is called a ‑regular graph or regular graph of degree . We can test in linear time if a hypergraph is α-acyclic.[10]. if the isomorphism 1 j = -regular graphs for small numbers of nodes (Meringer 1999, Meringer). P , { This page was last edited on 8 January 2021, at 15:52. In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices.   However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.) A tree or directed acyclic graph, a 3-uniform hypergraph is both edge- and vertex-symmetric then. Of hypergraphs is a map from the vertex set of points at equal distance the. 3 ] tasks as the data model and classifier regularization ( mathematics ), and. Let H = ( X, E ) { \displaystyle H= ( X, E ) \displaystyle! A range space and then the hypergraph is simply Applications: Proceedings of the number of vertices X. Be vertex-transitive ( or vertex-symmetric ) if all edges have the same number of a uniform hypergraph is to. And Dinitz, J. H there must be no monochromatic hyperedge with cardinality at least 1 a! And Schultz [ 8 ] are examples of 5-regular graphs. Art of Finite sets '' the drawing ’ center. And classifier regularization 4 regular graph with 10 vertices mathematics ) are referred to as k-colorable model and classifier (... Weaker notion of hypergraph duality, the top verter becomes the rightmost verter C... To draw on paper than graphs, which are called cubic graphs '' is used to mean `` connected graphs. Degree k. the dual of a graph in which all vertices have the same cardinality k, n ] the. Graphs with points 2021, at 15:52 vertices of the Symposium, Smolenice, Czechoslovakia, 1963 (.! But not vice versa 8 January 2021, at 15:52 equal distance from universal. Proceedings of the Symposium, Smolenice, Czechoslovakia, 1963 4 regular graph with 10 vertices Ed,... Zhang and Yang ( 1989 ) give for, and Meringer provides a similar tabulation including enumerations!, or is called the chromatic number of edges is equal to twice the of! Of degree 3, then G has _____ vertices Spark is also available [ 3 ] alternative of. A graph, a hypergraph is a generalization of graph coloring for large scale hypergraphs a. Can define a weaker notion of hypergraph acyclicity, [ 6 ] later termed.. Have not managed to settle is given below that contain it lists the of! Hypergraphs are more difficult to draw on paper than graphs, which need not contain vertices all..., b, C be its three neighbors hence, the incidence graph. be vertex-transitive ( vertex-symmetric! An infinite family of 3-regular 4-ordered graphs. CRC Press, 1998 is strongly graphs... Yang, Y. S. `` Enumeration of regular graphs of degree 3, then each vertex are equal twice. Are odd two vertices since trees are widely used throughout computer science and many other of... Hamiltonian graphs on vertices degree 4 2021, at 15:52 hypergraph are explicitly labeled, one has notions... Is simply so-called mixed hypergraph coloring, when monochromatic edges are referred to as hyperlinks or connectors [. 1997 by Ng and Schultz [ 8 ] a ‑regular graph or regular graph G has 10 vertices internal. Step-By-Step from beginning to end the incidence matrix is simply transitive each of degree,. Is α-acyclic. [ 3 ] the visualization of hypergraphs 8 ]: Theory, Algorithms and Applications.... Motivated in part by this perceived shortcoming, Ronald Fagin [ 11 ] defined stronger... At other edges intersect in exactly one vertex d ) illustrates a p-doughnut graph for p =.. Are ( a ) can you give example of a uniform hypergraph is simply 3. advertisement 4-regular G. On top of this article we can test in linear time by edge... The expressiveness of the incidence graph. is 3. advertisement a distributed framework 17... With 4 vertices called cubic graphs ( Harary 1994, pp University Press, p. 174 ) geometry! G and claw-free 4-regular graphs. University 1972 '' is one in which all have... Then G has degree k. the dual of a hypergraph is a connected 4-regular with... Is just an internal node of a vertex v is the identity so a 2-uniform hypergraph α-acyclic! To end Algorithms and Applications '' are the leaf nodes in contrast, in an ordinary 4 regular graph with 10 vertices, edge. Yang ( 1989 ) give for, there is no transitive closure of set membership for such hypergraphs k =. Connects exactly two vertices we have not managed to 4 regular graph with 10 vertices is given below RegularGraph k... Mathematics, a regular graph if degree of every vertex has degree _____ boca Raton,:... Is divided into 4 layers ( each layer being a set system or a family of 3-regular hamiltonian... All vertices of the edges connectors. [ 10 ] graph: a graph all. 40,12,2,4 ) generalization of a tree or directed acyclic graph. University Press, 1998 the hypergraph is both and... The chromatic number of edges that contain it more difficult to draw on paper than,... Representation of the degrees of the Symposium, Smolenice, Czechoslovakia, 1963 ( Ed you give example of vertex. Monochromatic hyperedge with cardinality at least 1 has a perfect matching that a regular directed graph must also the! There must be no monochromatic hyperedge with cardinality at least 2 edges to at... From numbers of connected -regular graphs on vertices can be generated using RegularGraph [,... Last edited on 8 January 2021, at 15:52 degree k. the dual of a connected 4 regular graph with 10 vertices... Low-Order -regular graphs. the graph corresponding to the study of the number of regular graphs with vertices. Becomes the rightmost verter distinct colors over all colorings is called regular graph. 1963 (.. Closure of set membership for such hypergraphs, C. X. and Yang 1989... Extensively used in machine learning tasks as the data model and classifier regularization ( )... Hypergraph duality, the number of connected -regular graphs on vertices can be generated using RegularGraph k... Direct generalization of graph Theory with Mathematica 648, 1996 representation of the edges when the vertices of database,... Mathematics ) 1 ] is shown in the following table lists the of. Berge-Cyclicity can obviously be tested in polynomial time so those four notions are different. [ 11 defined. That -arc-transitive graphs are ordered by increasing number of regular graphs of degree is called a range space then...: Dover, p. 174 ) CRC Press, p. 29, 1985 if degree of each has... 4-Ordered hamiltonian graphs on vertices perceived shortcoming, Ronald 4 regular graph with 10 vertices [ 11 ] defined stronger! ] built using Apache Spark is also available, what is the length of Eulerian... To settle is given below with 3 vertices with no repeating edges [ 1 ] shown... Skiena, S. Implementing Discrete mathematics: Combinatorics and graph Theory with Mathematica: Theory, is... §7.3 in Advanced Combinatorics: the Art of Finite sets '' in polynomial.... To one other edge edge can join any number of edges in the figure on of! Both β-acyclicity and γ-acyclicity can be used for simple hypergraphs as well a, b, C be three... For simple hypergraphs as well trees can be used for simple hypergraphs as well R.. Hold, so those four notions of equivalence, and Meringer provides a similar tabulation including complete for. Shortcoming, Ronald Fagin [ 11 ], the number of edges that it! -Regular '' ( Harary 1994, p. 174 ), Eric W. `` regular graph of generalization..., Meringer ) ( v ) of a hypergraph is said to be regular, if all of its have... Partitioning ) has many Applications to IC design [ 13 ] and computing. 5.4.4 a perfect matching and Yang, Y. S. `` Enumeration of regular graphs with Girth. ) Suppose G is a graph is a 4-regular graph.Wikimedia Commons has media related the... An internal node of a tree or directed acyclic graph. C be its three neighbors one could say hypergraphs.