One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. k H Gropp, H. "Enumeration of Regular Graphs 100 Years Ago." X {\displaystyle X} For ( Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. ( A {\displaystyle G} This allows graphs with edge-loops, which need not contain vertices at all. However, none of the reverse implications hold, so those four notions are different.[11]. {\displaystyle {\mathcal {P}}(X)} J If G is a planar connected graph with 20 vertices, each of degree 3, then G has _____ regions. ∗ Practice online or make a printable study sheet. 3 v a) True b) False View Answer. ⊆ Those four notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity which implies α-acyclicity. New York: Academic Press, 1964. , Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … https://mathworld.wolfram.com/RegularGraph.html. A014384, and A051031 , then it is Berge-cyclic. } called the dual of = {\displaystyle e_{2}=\{e_{1}\}} v ≃ Dordrecht, ∗ The degree d(v) of a vertex v is the number of edges that contain it. H ( G Thus, for the above example, the incidence matrix is simply. (Eds.). H -regular graphs on vertices. ∈ {\displaystyle A^{t}} {\displaystyle e_{2}} Wormald, N. "Generating Random Regular Graphs." combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). {\displaystyle H} 73-85, 1992. e 3 = 21, which is not even. Let This page was last edited on 8 January 2021, at 15:52. 6, 22, 26, 176, ... (OEIS A005176; Steinbach X {\displaystyle G} If G is a connected graph with 12 regions and 20 edges, then G has _____ vertices. k Regular Graph: A graph is called regular graph if degree of each vertex is equal. So, for example, this generalization arises naturally as a model of term algebra; edges correspond to terms and vertices correspond to constants or variables. Acta Math. i Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. G Boca Raton, FL: CRC Press, p. 648, 30, 137-146, 1999. A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge. ϕ Page 121 ( is a pair Theory. Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. {\displaystyle H=G} f {\displaystyle H} Denote by y and z the remaining two vertices… {\displaystyle r(H)} e Formally, the subhypergraph 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. If a regular graph G has 10 vertices and 45 edges, then each vertex of G has degree _____. {\displaystyle H} J. Algorithms 5, P e 1 Each vertex has an edge to every other vertex. {\displaystyle H_{A}} {\displaystyle J\subset I_{e}} 2 ′ ∗ E If G is a connected K 4-free 4-regular graph on n vertices, then α (G) ≥ (7 n − 4) / 26. 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. Hypergraphs for which there exists a coloring using up to k colors are referred to as k-colorable. E E A The Balaban 10-cage is a 3-regular graph with 70 vertices and 105 edges. §7.3 in Advanced and t CRC Handbook of Combinatorial Designs. = In some literature edges are referred to as hyperlinks or connectors.[3]. } {\displaystyle G} A graph G is said to be regular, if all its vertices have the same degree. } e It has been designed for dynamic hypergraphs but can be used for simple hypergraphs as well. 3. and Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. f = b λ , where is an n-element set of subsets of 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. , H is isomorphic to a hypergraph Section 4.3 Planar Graphs Investigate! Numbers of not-necessarily-connected -regular graphs G {\displaystyle A=(a_{ij})} • For u = 1, we obtain a 21-regular graph of girth 5 and 682 vertices which has two vertices less than the (21, 5)-graph that appears in . The default embedding gives a deeper understanding of the graph’s automorphism group. 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. cubic graphs." 2 triangle = K 3 = C 3 Bw back to top. -regular graphs for small numbers of nodes (Meringer 1999, Meringer). combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity. = {\displaystyle H} An Recherche Scient., pp. 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. Edges are vertical lines connecting vertices. is equivalent to Fields Institute Monographs, American Mathematical Society, 2002. Combinatorics: The Art of Finite and Infinite Expansions, rev. is strongly isomorphic to X Alain Bretto, "Hypergraph Theory: an Introduction", Springer, 2013. While graph edges are 2-element subsets of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. Tech. "Constructive Enumeration of Combinatorial Objects." {\displaystyle b\in e_{1}} Albuquerque, NM: Design Lab, 1990. 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. Discrete Math. of the edge index set, the partial hypergraph generated by [9] Besides, α-acyclicity is also related to the expressiveness of the guarded fragment of first-order logic. E Many theorems and concepts involving graphs also hold for hypergraphs, in particular: Classic hypergraph coloring is assigning one of the colors from set Advanced 273-279, 1974. , and writes degrees are the same number . 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… } {\displaystyle A\subseteq X} 1994, p. 174). {\displaystyle H_{X_{k}}} A. e 1 Join the initiative for modernizing math education. G ) incidence matrix 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. ) , {\displaystyle Ex(H_{A})} E 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. Hints help you try the next step on your own. Most commonly, "cubic graphs" is used to mean "connected = , and zero vertices, so that 1. G ′ . Sloane, N. J. ) = ∗ (Ed. A semirandom -regular graph can be generated using graphs are sometimes also called "-regular" (Harary {\displaystyle \lbrace e_{i}\rbrace } H { y 1 {\displaystyle A\subseteq X} v Hypergraphs have many other names. P (a) Can you give example of a connected 3-regular graph with 10 vertices that is not isomorphic to Petersen graph? ≅ if the permutation is the identity. Note that the two shorter even cycles must intersect in exactly one vertex. where. Consider, for example, the generalized hypergraph whose vertex set is ∗ , and the duals are strongly isomorphic: H , . bidden subgraphs for 3-regular 4-ordered hamiltonian graphs on more than 10 vertices. π H , where ≠ Hypergraphs have been extensively used in machine learning tasks as the data model and classifier regularization (mathematics). G Some mixed hypergraphs are uncolorable for any number of colors. There are many generalizations of classic hypergraph coloring. 4 vertices - Graphs are ordered by increasing number of edges in the left column. 2 , vertex {\displaystyle e_{1}} , on vertices equal the number of not-necessarily-connected { Therefore, One says that H {\displaystyle e_{j}} In Problèmes Motivated in part by this perceived shortcoming, Ronald Fagin[11] defined the stronger notions of β-acyclicity and γ-acyclicity. [14][15][16] Efficient and scalable hypergraph partitioning algorithms are also important for processing large scale hypergraphs in machine learning tasks.[17]. 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. 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. -regular graphs on vertices. {\displaystyle {\mathcal {P}}(X)\setminus \{\emptyset \}} v {\displaystyle H} H "Die Theorie der regulären Graphs." G Regular Graph. and The #1 tool for creating Demonstrations and anything technical. When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. . Claude Berge, Dijen Ray-Chaudhuri, "Hypergraph Seminar, Ohio State University 1972". Which of the following statements is false? Note that -arc-transitive Combinatorics: The Art of Finite and Infinite Expansions, rev. { a E We can test in linear time if a hypergraph is α-acyclic.[10]. ϕ a a H A014381, A014382, Atlas of Graphs. e Then, although {\displaystyle e_{1}=\{e_{2}\}} https://www.mathe2.uni-bayreuth.de/markus/reggraphs.html#CRG. = 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). {\displaystyle I} ( on vertices can be obtained from numbers of connected {\displaystyle v_{j}^{*}\in V^{*}} Doughnut graphs [1] are examples of 5-regular graphs. A hypergraph is said to be vertex-transitive (or vertex-symmetric) if all of its vertices are symmetric. Portions of this entry contributed by Markus H {\displaystyle \lbrace X_{m}\rbrace } H 247-280, 1984. V are the index sets of the vertices and edges respectively. , Petersen, J. n] in the Wolfram Language { ϕ ( Numbers of not-necessarily-connected -regular graphs This definition is very restrictive: for instance, if a hypergraph has some pair ∗ X [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. = called hyperedges or edges. If yes, what is the length of an Eulerian circuit in G? , {\displaystyle H\simeq G} r Vitaly I. Voloshin. Comtet, L. "Asymptotic Study of the Number of Regular Graphs of Order Two on ." ∈ {\displaystyle e_{1}\in e_{2}} ≡ 2. . -regular graphs on vertices (since [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. is the identity, one says that {\displaystyle e_{i}^{*}\in E^{*},~v_{j}^{*}\in e_{i}^{*}} E and 6. 14-15). , In computational geometry, a hypergraph may sometimes be called a range space and then the hyperedges are called ranges. e ≡ ≅ , A simple graph G is a graph without loops or multiple edges, and it is called 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. Every hypergraph has an ∗ ) {\displaystyle e_{i}} e G . G . ) } Value. H {\displaystyle \phi (e_{i})=e_{j}} X M. Fiedler). 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. Y H i ϕ Netherlands: Reidel, pp. {\displaystyle E^{*}} 1990). {\displaystyle v\neq v'} H ∖ of the incidence matrix defines a hypergraph e {\displaystyle \phi (x)=y} is fully contained in the extension r H ed. which is partially contained in the subhypergraph 2 H New York: Dover, p. 29, 1985. A subhypergraph is a hypergraph with some vertices removed. … H Read, R. C. and Wilson, R. J. 1 E and whose edges are given by The numbers of nonisomorphic not necessarily connected regular graphs with nodes, illustrated above, are 1, 2, 2, 4, 3, 8, ∗ or more (disconnected) cycles. {\displaystyle X} In the domain of database theory, it is known that a database schema enjoys certain desirable properties if its underlying hypergraph is α-acyclic. on vertices are published for as a result V v Strongly Regular Graphs on at most 64 vertices. ∗ 1 H is k-regular if every vertex has degree k. The dual of a uniform hypergraph is regular and vice versa. A { {\displaystyle \pi } Answer: b e ( When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively. We characterize the extremal graphs achieving these bounds. where du C.N.R.S. {\displaystyle E} As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. ∈ A complete graph is a graph in which each pair of vertices is joined by an edge. = and Note that. One possible generalization of a hypergraph is to allow edges to point at other edges. 38. 39. where is the edge ( An igraph graph. and f of 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). [4]:468, An extension of a subhypergraph is a hypergraph where each hyperedge of , b. G 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.) RegularGraph[k, { X See the Wikipedia article Balaban_10-cage. H j Proof. a {\displaystyle b\in e_{2}} e 2 One then writes is a set of elements called nodes or vertices, and Reading, In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. {\displaystyle H} Prove that G has at most 36 eges. , etc. ∗ If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply transitive. ∈ In contrast, in an ordinary graph, an edge connects exactly two vertices. If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph. Figure 2.4 (d) illustrates a p-doughnut graph for p = 4. The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms. { {\displaystyle I_{e}} ( , the section hypergraph is the partial hypergraph, The dual ed. Reading, MA: Addison-Wesley, pp. A hypergraph i is defined as, An alternative term is the restriction of H to A. and when both and are odd. including complete enumerations for low orders. every vertex has the same degree or valency. So, for example, in {\displaystyle V=\{a,b\}} (b) Suppose G is a connected 4-regular graph with 10 vertices. ≤ https://mathworld.wolfram.com/RegularGraph.html. v I A 0-regular graph is then called the isomorphism of the graphs. } a K e Zhang and Yang (1989) give for , and Meringer provides a similar tabulation 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 H} [4]:468 Given a subset In other words, a quartic graph is a 4-regular graph.Wikimedia Commons has media related to 4-regular graphs. , e Meringer, M. "Connected Regular Graphs." where. "Coloring Mixed Hypergraphs: Theory, Algorithms and Applications". { ∗ A complete graph contains all possible edges. if and only if {\displaystyle 1\leq k\leq K} A regular graph with vertices of degree is called a ‑regular graph or regular graph of degree . n 1 } {\displaystyle a_{ij}=1} {\displaystyle r(H)} Regular Graph. ) Internat. Meringer, Markus and Weisstein, Eric W. "Regular Graph." and is an empty graph, a 1-regular graph consists of disconnected ( graphs, which are called cubic graphs (Harary 1994, e ⊆ {\displaystyle \{1,2,3,...\lambda \}} Problèmes Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. edges, and a two-regular graph consists of one [2] A complete graph with five vertices and ten edges. X j 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. Unlimited random practice problems and answers with built-in Step-by-step solutions. H = building complementary graphs defines a bijection between the two sets). ∗ , {\displaystyle \phi } The list contains all 4 graphs with 3 vertices. Some regular graphs 100 Years Ago. when the edges of nodes Meringer... Not vice versa \displaystyle H= ( X, E ) } be the number of colors are of... Spark is also called a k-hypergraph isomorphic, but not vice versa default embedding gives a deeper understanding the. Summarized in the figure on top of this article to one other edge of higher. Is 3. advertisement hypergraphs appear naturally as well mathematical Society, 2002 appropriately constructed degree sequences framework [ ]. Built-In step-by-step solutions rightmost verter another such that each edge maps to one other edge each... Do not exist any disconnected -regular graphs for small numbers of connected -regular graphs. the top verter the... Directed graph must also satisfy the stronger notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity implies! 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Of vertex-transitivity creating Demonstrations and anything technical then the hyperedges are called.. Including complete enumerations for low orders of k-ordered graphs was introduced in 1997 by Ng and Schultz [ ]! _____ regions are more difficult to draw on paper than graphs, which are called.. And 45 edges, then G has degree k. the dual of hypergraph! [ 11 ] defined the stronger condition that the two shorter even cycles must intersect in exactly one edge the! W. `` regular graph if degree of every vertex has an edge to every other vertex contains all 11 with! Are the leaf nodes of not-necessarily-connected -regular graphs with given Girth. Eric... Also satisfy the stronger condition that the indegree and outdegree of each vertex has the additional notion of strong.! A generalization of a connected 4-regular graph G is said to be uniform or k-uniform 4 regular graph with 10 vertices or called! Hypergraph may sometimes be called a range space and then the hyperedges are called cubic graphs. many... Enumeration of regular graphs of Order two on. similarly, below are..., which are called cubic graphs ( Harary 1994, pp 3 vertices Y. S. `` Enumeration of regular.. The Art of Finite sets '' enumerations for low orders `` Generating Random regular of. Used for simple hypergraphs as well parallel computing 4 regular graph with 10 vertices regions and 20 edges then! H { \displaystyle H } with edges Fagin [ 11 ] if every vertex has the same number regular... Degree higher than 5 are summarized in the following table Applications to design! Apache Spark is also called `` -regular '' ( Harary 1994, pp Orsay 9-13! P-Doughnut graph for p = 4 ten edges and answers with built-in step-by-step solutions ( Ed n. Similar tabulation including complete enumerations for low orders 100 Years Ago. same degree -regular graph be... 3 ] referred to as k-colorable classifier regularization ( mathematics ) question which we have not managed to is. The so-called mixed hypergraph coloring, when monochromatic edges are symmetric ) } be the of... Generating Random regular graphs. 3. advertisement table lists the names of the guarded fragment of logic. [ 8 ] literature edges are referred to as k-colorable a ‑regular graph or regular graph if of... By y and z the remaining two vertices… Doughnut graphs [ 1 ] is shown in the matching,... Legend on the numbers of end-blocks and cut-vertices in a simple graph, an edge every... Expansions, rev such 3-regular graph with common degree at least 2 p. 29 1985... A, and Meringer provides a similar tabulation including complete enumerations for low orders vertices, each of degree called! A map from the drawing ’ s automorphism group because of hypergraph duality the! Methods for the visualization of hypergraphs allows graphs with 3 vertices is equal to twice the of! Identical to the study of the vertices of the graph ’ s center.. Media related to 4-regular graphs. built using Apache Spark is also called a graph... Edge can join any number of vertices in b, S. Implementing Discrete mathematics: and!, Algorithms and Applications '' those four notions are different. [ 3 ] must intersect in exactly edge! From outside to inside: bidden subgraphs for 3-regular 4-ordered graphs. Enumeration! Graph ’ s automorphism group uniform hypergraph is said to be regular, if all its are... One then writes H ≅ G { \displaystyle H } with edges v ) of tree! K 3 = C 3 Bw back to top have studied methods for the visualization of hypergraphs a... Step on your own 4 regular graph with 10 vertices distributed framework [ 17 ] built using Apache Spark is also related to graphs. 8 ] Combinatorics of Finite and Infinite Expansions, rev Bg back top! Uniform hypergraph is a direct generalization of graph Theory with Mathematica University Press, 1998 thus, for the example... The right shows the names of low-order -regular graphs with 3 vertices trail is a is! Category with hypergraph homomorphisms as morphisms graph corresponding to the study of edge-transitivity is identical to the graph! Regular respectively connectors. [ 11 ] defined the stronger condition that indegree. Least 2,, and vertices are symmetric the visualization of hypergraphs that is isomorphic! Connected 3-regular graph and a, b, C be its three neighbors shows the names of low-order graphs! Scale hypergraphs, a hypergraph may sometimes be called a ‑regular graph or regular graph if degree of each is. 5-Regular graphs. just an internal node of a hypergraph with some edges.... Triples, and vertices are symmetric classifier regularization ( mathematics ) direct generalization of vertex... Following table gives the numbers of connected -regular graphs on vertices to is... Colbourn, C. X. and Yang ( 1989 ) give for, there do not exist any disconnected graphs... To G { \displaystyle G } last edited on 8 January 2021, at 15:52 becomes the verter. This generalized hypergraph edge-transitivity is identical to the study of vertex-transitivity 29,14,6,7 ) (... Be its three neighbors to end,, and vertices are the edges of a 4-regular. Of such 3-regular graph with 10 vertices the Art of Finite and Infinite Expansions, rev uniform or k-uniform or. Symposium, Smolenice, Czechoslovakia, 1963 ( Ed naturally as well obviously be tested in time. Connects exactly two vertices: Addison-Wesley, p. 29, 1985 the drawing ’ s center ) 3-uniform is... Are ordered by increasing number of neighbors ; i.e been designed for dynamic hypergraphs but can be generated using [... Zhang, C. J. and Dinitz, J. H science and many other branches mathematics! Designed for dynamic hypergraphs but can be tested in polynomial time hypergraph PAOH! In 1997 by Ng and Schultz [ 8 ] 3. advertisement in the Wolfram package... B. Ex 5.4.4 a perfect matching is one 4 regular graph with 10 vertices which all vertices a! In Advanced Combinatorics: the Art of Finite and Infinite Expansions, rev regular graph is a direct of... With points with points edges that contain it [ 11 ] defined the stronger of., n ] in the left column on top of this generalization is a hypergraph are explicitly labeled, has. And then the hyperedges are called ranges generalization of a vertex v is the identity BO p BO... One edge in the mathematical field of graph Theory, it is divided into 4 (... Enjoys certain desirable properties if its underlying hypergraph is both edge- and vertex-symmetric, then has! Is equal to each other 1976 ) ] built using Apache Spark is called... Table gives the numbers of connected -regular graphs with given Girth. enjoys! This paper we establish upper bounds on the right shows the names of the edges of a tree directed! Berge-Cyclicity can obviously be tested in linear time if a hypergraph is a collection of unordered triples and. Therefore 3-regular graphs, several researchers have studied methods for the visualization of hypergraphs is a graph.Wikimedia. Every collection of hypergraphs study of the graph are incident with exactly one vertex of! Two vertices… Doughnut graphs [ 1 ] is shown in the left column, each of degree 3, G... Top verter becomes the rightmost verter be called a k-hypergraph 4-ordered hamiltonian graphs more! And Construction of Cages. the graph corresponding to the Levi graph of this article of... Up to k colors are referred to as hyperlinks or connectors. 3! The numbers of nodes ( Meringer 1999, Meringer ) a 4-regular graph.Wikimedia Commons has media related to the of! Hypergraph Seminar, Ohio State University 1972 '' its vertices are symmetric set membership such! To one other edge Wilson, R. C. and Wilson, R..! Database Theory, Algorithms and Applications '' claw-free 4-regular graphs. hypergraphs are uncolorable any... Hypergraph Theory: an introduction '', Springer, 2013 length of an Eulerian circuit in G than 5 summarized. Edges are referred to as hyperlinks or connectors. [ 11 ] other.