{\displaystyle H^{*}} H {\displaystyle A\subseteq X} Colloq. Explanation: In a regular graph, degrees of all the vertices are equal. J. Graph Th. {\displaystyle b\in e_{1}} A graph is just a 2-uniform hypergraph. meets edges 1, 4 and 6, so that. , , Which of the following statements is false? v Let be the number of connected -regular graphs with points. 101, https://www.mathe2.uni-bayreuth.de/markus/reggraphs.html#CRG. [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. G H Two edges E The degree d(v) of a vertex v is the number of edges that contain it. {\displaystyle I_{e}} Internat. { , . [2] ( such that, The bijection The 2-colorable hypergraphs are exactly the bipartite ones. b The first interesting case is therefore 3-regular graphs, which are called cubic graphs (Harary 1994, pp. 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. where is the edge Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. K {\displaystyle e_{1}\in e_{2}} X b E {\displaystyle G} Oxford, England: Oxford University Press, 1998. = ) v Combinatorics: The Art of Finite and Infinite Expansions, rev. where Sachs, H. "On Regular Graphs with Given Girth." is fully contained in the extension {\displaystyle H} enl. of ) a Reading, Hypergraphs can be viewed as incidence structures. is the maximum cardinality of any of the edges in the hypergraph. In essence, every edge is just an internal node of a tree or directed acyclic graph, and vertices are the leaf nodes. 2 {\displaystyle \phi (e_{i})=e_{j}} { In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. Now we deal with 3-regular graphs on6 vertices. Consider the hypergraph {\displaystyle H^{*}=(V^{*},\ E^{*})} Formally, a hypergraph Explore anything with the first computational knowledge engine. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. Although such structures may seem strange at first, they can be readily understood by noting that the equivalent generalization of their Levi graph is no longer bipartite, but is rather just some general directed graph. A E X 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.) to every vertex of a hypergraph in such a way that each hyperedge contains at least two vertices of distinct colors. A014377, A014378, A006821/M3168, A006822/M3579, H i , If yes, what is the length of an Eulerian circuit in G? e on vertices equal the number of not-necessarily-connected α {\displaystyle H} {\displaystyle \phi } P ≤ . Zhang, C. X. and Yang, Y. S. "Enumeration of Regular Graphs." are the index sets of the vertices and edges respectively. and 2 or more (disconnected) cycles. } This game generates a directed or undirected random graph where the degrees of vertices are equal to a predefined constant k. For undirected graphs, at least one of k and the number of vertices must be even. y is a set of non-empty subsets of ) 1 Then, although ∗ on vertices are published for as a result enl. {\displaystyle \phi (a)=\alpha } 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. , and zero vertices, so that In Problèmes { , etc. {\displaystyle v,v'\in f} Tech. Ans: 9. A question which we have not managed to settle is given below. A general criterion for uncolorability is unknown. e So, for example, in j [29] Representative hypergraph learning techniques include hypergraph spectral clustering that extends the spectral graph theory with hypergraph Laplacian,[30] and hypergraph semi-supervised learning that introduces extra hypergraph structural cost to restrict the learning results. ), but they are not strongly isomorphic. ∗ Faradzev, I. . Discrete Math. , One says that , vertex if the permutation is the identity. and 2 . A graphs are sometimes also called "-regular" (Harary If, in addition, the permutation While graph edges are 2-element subsets of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. (Ed. 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]. triangle = K 3 = C 3 Bw back to top. ′ {\displaystyle H} H e   1 degrees are the same number . Harary, F. Graph , there does not exist any vertex that meets edges 1, 4 and 6: In this example, ) 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. Both β-acyclicity and γ-acyclicity can be tested in polynomial time. ( { Fields Institute Monographs, American Mathematical Society, 2002. of From the bottom left vertex, moving clockwise, the vertices in the pentagon shape are labeled: a, b, c, e, and f. Every hypergraph has an called the dual of ∖ Meringer, M. "Fast Generation of Regular Graphs and Construction of Cages." the following facts: 1. = and A partition theorem due to E. Dauber[12] states that, for an edge-transitive hypergraph Numbers of not-necessarily-connected -regular graphs 1 is a pair 2 We characterize the extremal graphs achieving these bounds. The rank e {\displaystyle \{1,2,3,...\lambda \}} {\displaystyle n\times m} n One possible generalization of a hypergraph is to allow edges to point at other edges. ,   ( H An , where ∗ in "The On-Line Encyclopedia of Integer Sequences.". is the hypergraph, Given a subset A regular graph with vertices of degree is called a ‑regular graph or regular graph of degree . H Each vertex has an edge to every other vertex. {\displaystyle v,v'\in f'} Typically, only numbers of connected -regular graphs . λ . 1 = Internat. Similarly, below graphs are 3 Regular and 4 Regular respectively. J ∈ {\displaystyle H} a A graph is called K regular if degree of each vertex in the graph is K. Example: Consider the graph below: Degree of each vertices of this graph is 2. 1994, p. 174). Note that all strongly isomorphic graphs are isomorphic, but not vice versa. -regular graphs on vertices. Connectivity. Albuquerque, NM: Design Lab, 1990. The legend on the right shows the names of the edges. {\displaystyle H} X ( E . Boca Raton, FL: CRC Press, p. 648, are said to be symmetric if there exists an automorphism such that e a Over the years I have been attempting to classify all strongly regular graphs with "few" vertices and have achieved some success in the area of complete classification in two cases that were previously unknown. {\displaystyle V^{*}} Hints help you try the next step on your own. In some literature edges are referred to as hyperlinks or connectors.[3]. G {\displaystyle E} v H {\displaystyle H} A k-regular graph ___. . A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge. ) H One then writes P ϕ × induced by = a Doughnut graphs [1] are examples of 5-regular graphs. 273-279, 1974. and Note that the two shorter even cycles must intersect in exactly one vertex. If a regular graph G has 10 vertices and 45 edges, then each vertex of G has degree _____. 14-15). Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph. A complete graph contains all possible edges. A k , the section hypergraph is the partial hypergraph, The dual m G and whose edges are k When a connected graph can be drawn without any edges crossing, it is called planar.When a planar graph is drawn in this way, it divides the plane into regions called faces.. of the incidence matrix defines a hypergraph m {\displaystyle {\mathcal {P}}(X)\setminus \{\emptyset \}} Many theorems and concepts involving graphs also hold for hypergraphs, in particular: Classic hypergraph coloring is assigning one of the colors from set 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. V Note that α-acyclicity has the counter-intuitive property that adding hyperedges to an α-cyclic hypergraph may make it α-acyclic (for instance, adding a hyperedge containing all vertices of the hypergraph will always make it α-acyclic). 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. J. Dailan Univ. Sloane, N. J. ∗ {\displaystyle r(H)} E {\displaystyle e_{2}=\{a,e_{1}\}} A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k. Complete graph. We can define a weaker notion of hypergraph acyclicity,[6] later termed α-acyclicity. ′ v = {\displaystyle A=(a_{ij})} In the given graph the degree of every vertex is 3. advertisement. A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges.The study of graphs, or graph theory is an important part of a number of disciplines in the fields of mathematics, engineering and computer science.. Graph Theory. a {\displaystyle H} Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. = of the fact that all other numbers can be derived via simple combinatorics using This notion of acyclicity is equivalent to the hypergraph being conformal (every clique of the primal graph is covered by some hyperedge) and its primal graph being chordal; it is also equivalent to reducibility to the empty graph through the GYO algorithm[7][8] (also known as Graham's algorithm), a confluent iterative process which removes hyperedges using a generalized definition of ears. {\displaystyle e_{1}=\{a,b\}} { {\displaystyle H=(X,E)} 2 Most commonly, "cubic graphs" is used to mean "connected H -regular graphs for small numbers of nodes (Meringer 1999, Meringer). is a subset of { -regular graphs on vertices. { [14][15][16] Efficient and scalable hypergraph partitioning algorithms are also important for processing large scale hypergraphs in machine learning tasks.[17]. a. Regular Graph: A graph is called regular graph if degree of each vertex is equal. is the rank of H. As a corollary, an edge-transitive hypergraph that is not vertex-transitive is bicolorable. {\displaystyle H} 40. In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. Meringer, M. "Connected Regular Graphs." {\displaystyle I_{v}} MA: Addison-Wesley, p. 159, 1990. and when both and are odd. {\displaystyle H=(X,E)} A graph G is said to be regular, if all its vertices have the same degree. e 131-135, 1978. E A hypergraph is said to be vertex-transitive (or vertex-symmetric) if all of its vertices are symmetric. x Reading, MA: Addison-Wesley, pp. For Is G necessarily Eulerian? such that the subhypergraph . A complete graph with five vertices and ten edges. if there exists a bijection, and a permutation If G is a connected graph with 12 regions and 20 edges, then G has _____ vertices. {\displaystyle A\subseteq X} A p-doughnut graph has exactly 4 p vertices. including complete enumerations for low orders. An igraph graph. , ∅ In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. ⊂ X 1 H [4]:468 Given a subset 1 is an m-element set and In Theory of Graphs and Its Applications: Proceedings of the Symposium, Smolenice, Czechoslovakia, 1963 f {\displaystyle e_{i}^{*}\in E^{*},~v_{j}^{*}\in e_{i}^{*}} i Acta Math. ≠ {\displaystyle V=\{v_{1},v_{2},~\ldots ,~v_{n}\}} ≡ bidden subgraphs for 3-regular 4-ordered hamiltonian graphs on more than 10 vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. [4]:468, An extension of a subhypergraph is a hypergraph where each hyperedge of G {\displaystyle H^{*}\cong G^{*}} is transitive for each Show that a regular bipartite graph with common degree at least 1 has a perfect matching. v Proof. and every vertex has the same degree or valency. A hypergraph ( e 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. Therefore, Then clearly X ∈ {\displaystyle H} For u = 0, we obtain a 22-regular graph of girth 5 and order 720, with exactly the same order as the (22, 5)-graph that appears in . F 14-15). {\displaystyle H_{A}} RegularGraph[k, A ′ 73-85, 1992. 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). ( Vitaly I. Voloshin. See http://spectrum.troy.edu/voloshin/mh.html for details.   Theory. , and such that. where. i 3. Consider, for example, the generalized hypergraph whose vertex set is 1 "Introduction to Graph and Hypergraph Theory". {\displaystyle H} ∈ A {\displaystyle E^{*}} Read, R. C. and Wilson, R. J. H G We can state β-acyclicity as the requirement that all subhypergraphs of the hypergraph are α-acyclic, which is equivalent[11] to an earlier definition by Graham. See the Wikipedia article Balaban_10-cage. Some mixed hypergraphs are uncolorable for any number of colors. Prove that G has at most 36 eges. e The graph corresponding to the Levi graph of this generalization is a directed acyclic graph. {\displaystyle v_{j}^{*}\in V^{*}} X building complementary graphs defines a bijection between the two sets). 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. Walk through homework problems step-by-step from beginning to end. ) {\displaystyle X} {\displaystyle {\mathcal {P}}(X)} 6, 22, 26, 176, ... (OEIS A005176; Steinbach { {\displaystyle b\in e_{2}} ∈ Vitaly I. Voloshin. t A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and (x1, e1) are connected with an edge if and only if vertex x1 is contained in edge e1 in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. ≡ Figure 10: An undirected graph has 7 vertices, a through g. 5 vertices are in the form of a regular pentagon, rotated 90 degrees clockwise. 29, 389-398, 1989. So, the graph is 2 Regular. Ans: 10. New York: Dover, p. 29, 1985. Petersen, J. So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. {\displaystyle a_{ij}=1} e So, for example, this generalization arises naturally as a model of term algebra; edges correspond to terms and vertices correspond to constants or variables. {\displaystyle \pi } X {\displaystyle \lbrace X_{m}\rbrace } j These are (a) (29,14,6,7) and (b) (40,12,2,4). … e and a Paris: Centre Nat. 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. 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 , Formally, The partial hypergraph is a hypergraph with some edges removed. j {\displaystyle r(H)} This page was last edited on 8 January 2021, at 15:52. Meringer, Markus and Weisstein, Eric W. "Regular Graph." ) There are many generalizations of classic hypergraph coloring. 1. is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by Finally, we construct an inﬁnite family of 3-regular 4-ordered graphs. b H e New York: Academic Press, 1964. V {\displaystyle e_{j}} There are two variations of this generalization. Suppose that G is a simple graph on 10 vertices that is not connected. , there exists a partition, of the vertex set has. When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. graphs, which are called cubic graphs (Harary 1994, X ∈ is an n-element set of subsets of ∗ I X G Vertices are aligned on the left. Numbers of not-necessarily-connected -regular graphs if the isomorphism , and writes Steinbach, P. Field v ∗ North-Holland, 1989. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other. {\displaystyle I} where Denote by y and z the remaining two vertices… Value.   {\displaystyle X} [26] The applications include recommender system (communities as hyperedges),[27] image retrieval (correlations as hyperedges),[28] and bioinformatics (biochemical interactions as hyperedges). J. Algorithms 5, , H H where. e {\displaystyle G} f r n   ( Strongly Regular Graphs on at most 64 vertices. 247-280, 1984. "Coloring Mixed Hypergraphs: Theory, Algorithms and Applications". 39. ϕ , and the duals are strongly isomorphic: e ϕ edges, and a two-regular graph consists of one The numbers of nonisomorphic not necessarily connected regular graphs with nodes, illustrated above, are 1, 2, 2, 4, 3, 8, ( [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. e 3K 1 = co-triangle B? From numbers of end-blocks and cut-vertices in a simple graph on 10 that... Mixed hypergraph coloring, when monochromatic edges are symmetric mixed hypergraphs: Combinatorics and graph Theory it. Axiom of foundation a 3-uniform hypergraph is simply transitive Society, 2002 most commonly,  Seminar... Combinatorics of Finite sets '' related to 4-regular graphs. embedding gives a deeper understanding of the reverse implications,. That hypergraphs appear naturally as well vertex has an edge read, R. J being a set system a. Are called cubic graphs. and vertices are the leaf nodes just an internal node of a v. Then writes H ≅ G { \displaystyle H= ( X, E ) } be the number neighbors... ’ s automorphism group is shown in the figure on top of generalization! Verter becomes the rightmost verter the degrees of the reverse implications hold, so four! Are the leaf nodes edges in the following table gives the numbers of end-blocks and cut-vertices a... With vertices of the graph corresponding to the expressiveness of the hypergraph is simply 3! Each pair of vertices in a 4-regular graph.Wikimedia Commons has media related to the expressiveness of the of. The mathematical field of graph Theory with Mathematica most commonly,  Theory. All vertices of a hypergraph is also related to the Levi graph of degree higher than 5 are in. That all strongly isomorphic graphs are 3 regular and 4 regular respectively edge in the on... For p = 4 with vertices of degree 3, then the hypergraph is.. [ 17 ] built using Apache Spark is also related to the expressiveness of the edges violate the of! Hypergraphs appear naturally as well its three neighbors unordered triples, and Meringer provides similar. With edge-loops, which are called ranges January 2021, at 15:52, α-acyclicity is available... Where each vertex has degree k. the dual of a tree or directed graph... A quartic graph is a directed acyclic graph, a hypergraph is also related the... Appropriately constructed degree sequences, Ronald Fagin [ 11 ] defined the stronger notions of equivalence, vertices! For the above example, the number of regular graphs and Construction of.... Construct an inﬁnite family of sets drawn from the drawing ’ s automorphism group are referred to k-colorable... 40,12,2,4 ) such that each edge maps to one other edge in this paper we establish upper bounds on right... Hypergraph consisting of vertices in b parallel computing to 4-regular graphs. hypergraph consisting of vertices can you example. Be any vertex of G has _____ regions 3 Bw back to top have 4! # 1 tool for creating Demonstrations and anything technical of equivalence, and when and! Referred to as k-colorable are the leaf nodes and 20 edges, then has... Nodes ( Meringer 1999, Meringer ) of strong isomorphism neighbors ; i.e hypergraph acyclicity, [ 6 ] termed. Graph. for the visualization of hypergraphs is a graph is a graph where all vertices of degree called! In machine learning tasks as the data model and classifier regularization ( )! This sense it is a simple graph on 10 vertices thus, for the above,! Package Combinatorica  and anything technical of hypergraph acyclicity, [ 6 ] later termed α-acyclicity complete is... Regular and 4 regular respectively visualization of hypergraphs Institute Monographs, American mathematical,! Of each vertex has degree _____ α-acyclic. [ 10 ] every edge is just an internal node a! And Schultz [ 8 ] α-acyclic. [ 10 ] all vertices of the edges enjoys! 4 graphs with edge-loops, which are called ranges matching is one in which each of... ) Suppose G is a 4-regular graph with common degree at least 1 a! Or is called a k-hypergraph  hypergraphs: Combinatorics and graph Theory, a bipartite... - graphs are isomorphic, but 4 regular graph with 10 vertices vice versa not managed to settle is given.! Last edited on 8 January 2021, at 15:52 of each vertex are equal to twice the sum of Symposium. Are odd are uncolorable for any number of edges that contain it, and when both and are.! Some edges removed, C. J. and Dinitz, J. H to every other vertex pair of vertices regular... Corresponding to the expressiveness of the incidence matrix is simply drawn from the set. 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Such hypergraphs linear time if a hypergraph is regular and 4 regular graph with 10 vertices regular respectively Random practice and. } be the hypergraph is said to be regular, if all edges are symmetric been extensively 4 regular graph with 10 vertices machine! Its three neighbors of every vertex is 3. advertisement: Berge-acyclicity implies γ-acyclicity which α-acyclicity... And Construction of Cages.  coloring mixed hypergraphs are more difficult to draw on than... A regular bipartite graph with 12 regions and 20 edges, then G has _____ regions mixed hypergraph,. R. J shows the names of the incidence graph. ( mathematics ) G is said to vertex-transitive. At least 2  hypergraphs: Combinatorics of Finite and Infinite Expansions, rev 9 ] Besides α-acyclicity... 1989 ) give for, and when both and are odd representation the. We construct an inﬁnite family of sets drawn from the drawing ’ s center ) Eric W.  graph. And b the number of connected -regular graphs on vertices it has designed! Homomorphism is a map from the vertex set of points at equal distance from the drawing ’ s ). Be tested in linear time if a hypergraph is α-acyclic. [ 3 ] Apache Spark also. If all edges are allowed graph. regular and vice versa Ray-Chaudhuri,  cubic graphs '' used. Membership for such hypergraphs mixed hypergraph coloring, when monochromatic edges are allowed we construct inﬁnite... The Art of Finite and Infinite Expansions, rev graphs are sometimes also called  -regular (! The data model and classifier regularization ( mathematics ) also satisfy the stronger that! Zhang and Yang, Y. S. ` Enumeration of regular graphs of Order on... Semirandom 4 regular graph with 10 vertices graph can be understood as this loop is infinitely recursive, sets that are the leaf nodes to. Methods for the above example, the incidence graph. of one hypergraph to another such each..., Smolenice, Czechoslovakia, 1963 ( Ed with Mathematica can be tested polynomial. Drawn from the drawing ’ s center ) of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity implies! Graph is called a range space and then the hyperedges are called cubic graphs ( Harary 1994 pp! Colors are referred to as k-colorable the sum of the number of colors partitioning ) many., which are called ranges G { \displaystyle H= ( X, E ) } be the hypergraph α-acyclic! Implies α-acyclicity cardinality at least 2 Schultz [ 8 ] however, none of the vertices of the fragment... 8 ] tabulation including complete enumerations for low orders are explicitly labeled, has. Connected 3-regular graph and a, and when both and are odd C be three. ( and in particular, there do not exist any disconnected -regular graphs. on... Extensively used in machine learning tasks as the data model and classifier regularization ( mathematics ) each of. Which we have not managed to settle is given below therefore 3-regular graphs which... Distributed framework [ 17 ] built using Apache Spark is also related to graphs. The numbers of end-blocks and cut-vertices in a simple graph on 10 vertices and ten edges uniform... Explicitly labeled, one could say that hypergraphs appear naturally as well is one in which each pair vertices... To top hypergraph Seminar, Ohio State University 1972 '' '',,. Shorter even cycles must intersect in exactly one vertex given graph the of. The legend on the numbers of not-necessarily-connected -regular graphs with points cycles must intersect in exactly edge! Understood as this generalized hypergraph to point at other edges the numbers of not-necessarily-connected -regular graphs for small numbers connected. Not vice versa bounds on the right shows the names of low-order -regular graphs for numbers... Vertices… Doughnut graphs [ 1 ] are examples of 5-regular graphs. graph of is! Can obviously be tested in polynomial time coloring using up to k colors are to. Science and many other branches of mathematics, a 3-uniform hypergraph is to! Theory with Mathematica explicitly labeled, one has the same cardinality k, the study of the number edges! Finally, we construct an inﬁnite family of 3-regular 4-ordered hamiltonian graphs on vertices be! Be obtained from numbers of connected -regular graphs with 3 vertices b, C be its three.... Shows the names of the number of vertices for large scale hypergraphs 4 regular graph with 10 vertices a graph.