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field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. The triangle graph is also
Triangle_graph
Graph without triples of adjacent vertices
area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently
Triangle-free_graph
Triangle-free graph requiring four colors
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number
Grötzsch_graph
Directed graph with no directed cycles
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it
Directed_acyclic_graph
Planar graph used as counterexample
In graph theory, a nested triangles graph with n vertices is a planar graph formed from a sequence of n/3 triangles, by connecting pairs of corresponding
Nested_triangles_graph
Fractal composed of triangles
form an undirected graph, the Hanoi graph, that can be represented geometrically as the intersection graph of the set of triangles remaining after the
Sierpiński_triangle
Graph representing edges of another graph
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges
Line_graph
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
1970. It is the smallest graph that is triangle-free, 4-regular, and 4-chromatic. The Chvátal graph is triangle-free: its girth (the length of its shortest
Chvátal_graph
Topics referred to by the same term
The triangle graph in graph theory Triangle (musical instrument), in the percussion family Tri Angle (record label), in New York and London Triangle (band)
Triangle_(disambiguation)
Intersection graph of a chord diagram
In graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with
Circle_graph
Graph divided into two independent sets
required in the graph coloring problem. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node
Bipartite_graph
Barycentric plot on three variables
A ternary plot, ternary graph, triangle plot, simplex plot, or Gibbs triangle is a barycentric plot on three variables which sum to a constant. It graphically
Ternary_plot
Length of a shortest cycle contained in the graph
well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free. A cubic graph (all vertices have degree three) of girth g
Girth_(graph_theory)
Methodic assignment of colors to elements of a graph
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain
Graph_coloring
Trail in which only the first and last vertices are equal
directed graph with no directed cycles Forest, a cycle-free graph Line perfect graph, a graph in which every odd cycle is a triangle Perfect graph, a graph with
Cycle_(graph_theory)
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Cubic graph with 8 vertices and 12 edges
mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph. As a Möbius ladder
Wagner_graph
Geometric graph with unit edge lengths
complete graph on two vertices is a unit distance graph, as is the complete graph on three vertices (the triangle graph), but not the complete graph on four
Unit_distance_graph
Theorem in graph theory
The special case in which the subgraph is a triangle is known as the triangle removal lemma. The graph removal lemma can be used to prove Roth's theorem
Graph_removal_lemma
Derived graph of higher chromatic number
exist triangle-free graphs with arbitrarily large chromatic number. Let the n vertices of the given graph G be v1, v2, . . . , vn. The Mycielski graph μ(G)
Mycielskian
Prism with a 3-sided base
equilateral triangle faces on the planes of the triangular ends of the prism, and six more isosceles triangle faces. By giving rise to a graph, its dual
Triangular_prism
Graph with a median for each three vertices
In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a {\displaystyle a} , b {\displaystyle
Median_graph
Graph where every edge is in one triangle
In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the
Locally_linear_graph
On existence of a strongly regular graph
graph could exist with each edge in a unique triangle and each non-edge forming the diagonal of a unique quadrilateral. It is only known that graphs exist
Conway's_99-graph_problem
Writing paper with a grid
Isometric graph paper or 3D graph paper is a triangular graph paper which uses a series of three guidelines forming a 60° grid of small triangles. The triangles
Graph_paper
Solid with eight equal triangular faces
structure, the graph forms a topological surface called a Whitney triangulation. A regular octahedron is a polyhedron with eight equilateral triangles. Each vertex
Regular_octahedron
Planar graph with 5 nodes and 6 edges
induced subgraph. The triangle-free graphs are bowtie-free graphs, since every butterfly contains a triangle. In a k-vertex-connected graph, an edge is said
Butterfly_graph
In graph theory and theoretical computer science, the monochromatic triangle problem is an algorithmic problem on graphs, in which the goal is to partition
Monochromatic_triangle
Graph in which every two vertices are adjacent
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique
Complete_graph
Property of graphs that depends only on abstract structure
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations
Graph_property
Subgraph induced by all nodes linked to a given node of a graph
Turán graph is locally Turán. Every planar graph is locally outerplanar. However, not every locally outerplanar graph is planar. A graph is triangle-free
Neighbourhood_(graph_theory)
Graph that can be embedded in the plane
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect
Planar_graph
Infinite graph without small cliques
first of these graphs, G3, is also called the homogeneous triangle-free graph or the universal triangle-free graph. To construct these graphs, Henson orders
Henson_graph
One of two different regular graphs with 16 vertices
field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with
Clebsch_graph
Adjacent subset of an undirected graph
some clique contains at least one endpoint of every edge. A triangle-free graph is a graph that has no cliques other than its vertices and edges. Additionally
Clique_(graph_theory)
Every triangle-free planar graph is 3-colorable
In the mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors
Grötzsch's_theorem
Task of computing complete subgraphs
clique-finding problem is finding a triangle in a graph, or equivalently determining whether the graph is triangle-free. In a graph G with m edges, there may be
Clique_problem
Square matrix used to represent a graph or network
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether
Adjacency_matrix
Planar graph with 4 nodes and 5 edges
Hamiltonian graph. A graph is diamond-free if it has no diamond as an induced subgraph. The triangle-free graphs are diamond-free graphs, since every
Diamond_graph
Cubic graph with 10 vertices and 15 edges
Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle. If G is a 2-connected, r-regular graph with
Petersen_graph
Graph without four-vertex star subgraphs
Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. Claw-free graphs were initially studied
Claw-free_graph
Every graph has evenly many odd vertices
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges
Handshaking_lemma
Graph representing faces of another graph
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each
Dual_graph
Family of triangle-free circulant graphs
In graph theory, an Andrásfai graph is a triangle-free, circulant graph named after Béla Andrásfai. The Andrásfai graph And(n) for any natural number n
Andrásfai_graph
Structure-preserving correspondence between node-link graphs
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a
Graph_homomorphism
Archimedean solid with 8 faces
cubic graph, and connected cubic transitive graph. As a Wythoff construction, it is vertex transitive, and has 2 edge orbits of 6 (between triangles) and
Truncated_tetrahedron
Shape representing matchings in a graph
of incidence matrices: the triangle graph (a cycle of length 3), a square graph (a cycle of length 4), and the complete graph on 4 vertices. For every subset
Matching_polytope
Concept in extremal graph theory
since all Sidorenko graphs are bipartite graphs while there exist non-bipartite common graphs, as demonstrated below. The triangle graph K 3 {\displaystyle
Common_graph
Graph of triangles with a shared vertex
the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or n-fan) Fn is a planar, undirected graph with 2n + 1 vertices and
Friendship_graph
Topic in computer science
long time, the best known algorithm for testing whether a graph does not contain any triangle had a query complexity which is a tower function of poly(1/ε)
Property_testing
Bipartite graph where each node of 1st set is linked to all nodes of 2nd set
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first
Complete_bipartite_graph
Graph with edges of length one, able to be drawn without crossings
instance the three different graphs that can be made with three matchsticks are a claw, a triangle graph, and a three-edge path graph. Uniformity of edge lengths
Matchstick_graph
Convex polyhedron with 14 triangle faces
generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form
Triaugmented_triangular_prism
Strongly regular graph
Kneser graphs), these are the unique maximum independent sets in this graph. It is one of seven known triangle-free strongly regular graphs. Its graph spectrum
M22_graph
Partition of a simple polygon into triangles
P, one defines the graph G(TP) as the graph whose vertex set are the triangles of TP, two vertices (triangles) being adjacent if and only if they share
Polygon_triangulation
Measure of how connected and clustered a node is in its graph
{\displaystyle \lambda _{G}(v)} be the number of triangles on v ∈ V ( G ) {\displaystyle v\in V(G)} for undirected graph G {\displaystyle G} . That is, λ G ( v
Clustering_coefficient
Shape with three equal sides
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral
Equilateral_triangle
mathematical field of graph theory, the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant
Bull_graph
Concept in graph theory
In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers λ , μ ≥ 0
Strongly_regular_graph
Archimedean solid with 62 faces
pentagrammic prisms. In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron
Rhombicosidodecahedron
Bipartite, 3-regular undirected graph
nine-vertex graph is 6-regular, is the complement graph of the union of three disjoint triangle graphs, and is the complete tripartite graph K3,3,3. The
Pappus_graph
Branch of discrete mathematics
Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations
Combinatorics
Graph formed by touching unit circles
penny graphs that do still require four colors. Analogously, the degeneracy of every triangle-free penny graph is at most two. Every such graph contains
Penny_graph
Archimedean solid with 32 faces
identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it
Icosidodecahedron
Undirected cubic graph with 12 vertices and 18 edges
Tietze's graph. Tietze's graph may be formed from the Petersen graph by replacing one of its vertices with a triangle. Like the Tietze graph, the Petersen
Tietze's_graph
Graph of chess rook moves
one exception, the rook's graphs can be distinguished from all other graphs using only two properties: the numbers of triangles each edge belongs to, and
Rook's_graph
Non-crossing graph with vertices on outer face
every bounded face of a maximal outerplanar graph is a triangle. Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski's theorem
Outerplanar_graph
Graph able to be embedded on a torus
toroidal graph has chromatic number at most 7. The complete graph K7 provides an example of a toroidal graph with chromatic number 7. Any triangle-free toroidal
Toroidal_graph
Graph of numbers differing by a square
Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic
Paley_graph
Graph whose peripheral cycles are all triangles
disconnect the remaining graph. That is, they are the graphs in which every peripheral cycle is a triangle. In a maximal planar graph, or more generally in
Strangulated_graph
Subgraph of Delaunay triangulation
Urquhart graph of a set of points in the plane, named after Roderick B. Urquhart, is obtained by removing the longest edge from each triangle in the Delaunay
Urquhart_graph
Graph formed by subdivision of triangles
Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently
Apollonian_network
Archimedean solid with 38 faces
is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. Kepler first named it in Latin as cubus
Snub_cube
every edge belongs to a unique triangle. Equivalently it asks for the maximum number of edges in a balanced bipartite graph whose edges can be partitioned
Ruzsa–Szemerédi_problem
Graph representing connectivity between cliques of another graph
simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph. The simplex graph of
Simplex_graph
Franklin graph Frucht graph Goldner–Harary graph Golomb graph Grötzsch graph Harries graph Harries–Wong graph Herschel graph Hoffman graph Hofman Graph H(12
List_of_graphs
Pattern of states and moves in the Tower of Hanoi puzzle
In graph theory and recreational mathematics, the Hanoi graphs are undirected graphs whose vertices represent the possible states of the Tower of Hanoi
Hanoi_graph
Archimedean solid with 26 faces
rhombicuboctahedron is a polyhedron with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices
Rhombicuboctahedron
S2CID 119169562. Zbl 1218.05034.. Tuza, Zsolt (1990). "A conjecture on triangles of graphs". Graphs and Combinatorics. 6 (4): 373–380. doi:10.1007/BF01787705. MR 1092587
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Graph representing tangency between geometric objects
unit circles are called penny graphs. Representations as contact graphs of triangles, rectangles, squares, line segments, or circular arcs have also been
Contact_graph
Hierarchical clustering of graph edges
In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree
Branch-decomposition
Archimedean solid with 14 faces
excavated cube has 16 triangles, 12 squares, and 4 octagons. In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and
Truncated_cube
Method for finding minimum spanning trees
ensure that the created graph is indeed a forest, that is, it does not contain cycles. For example, consider a triangle graph with nodes {a,b,c} and all
Borůvka's_algorithm
Mathematical tree of cycles
triangles joined together at a single shared vertex, are triangular cacti. As well as being cactus graphs the triangular cacti are also block graphs and
Cactus_graph
Special type of Boolean function
co-occurrence graph as the conjunction of three variables, a triangle graph, but the three-vertex complete subgraph of this graph (the whole graph) forms a
Read-once_function
Graph whose embedding in a Euclidean space forms a regular tiling
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space R n {\displaystyle \mathbb {R}
Lattice_graph
Topics referred to by the same term
maximal planar supergraph of a graph G may be called a triangulation of G Triangulated graph, and a chordal completion of a graph G may be called a triangulation
Triangulation (disambiguation)
Triangulation_(disambiguation)
Triangulation method
general position corresponds to the dual graph of the Voronoi diagram for P. The circumcenters of Delaunay triangles are the vertices of the Voronoi diagram
Delaunay_triangulation
40824 edges (112 per vertex). Each edge is in a unique triangle (it is a locally linear graph) and each non-adjacent pair of vertices have exactly 20
Games_graph
the graph is planar and F indicates that the graph is not planar. Wikimedia Commons has media related to Graphs by number of vertices. See also Graph theory
List of graphs by edges and vertices
List_of_graphs_by_edges_and_vertices
Polyhedron made of equilateral triangles
142–144. Eppstein, D. (2021), "On Polyhedral Realization with Isosceles Triangles", Graphs and Combinatorics, 37 (4), Springer: 1247–1269, arXiv:2009.00116,
Deltahedron
Extremal graph theory bound on clique-free graph edges
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given
Turán's_theorem
Mathematical graph theorem
dual graph of a triangle mesh and connecting pairs of triangles that are not matched, one can decompose the mesh into cyclic strips of triangles. With
Petersen's_theorem
Operation on graphs
ΔY-transformation. Let G {\displaystyle G} be a graph (potentially a multigraph). Suppose G {\displaystyle G} contains a triangle Δ {\displaystyle \Delta } with vertices
YΔ-_and_ΔY-transformation
On short connecting nets with added points
term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually
Steiner_tree_problem
3-regular graph with no 3-edge-coloring
giving the name "snark" to these graphs, Gardner lists Tietze's graph, which contains a triangle, as being a snark. If a graph contains a four-vertex cycle
Snark_(graph_theory)
Statement in mathematical combinatorics
colouring of a complete graph using 3 colours, red, green and blue. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex v
Ramsey's_theorem
Polyhedron with 8 triangles and 6 squares
the 8 triangles split into 2 face classes. The square symmetry is reduced to 2-fold. The skeleton of a cuboctahedron may be represented as the graph, one
Cuboctahedron
Graph with same nodes as but complementary connections to another
automorphism group of a graph is the automorphism group of its complement. The complement of every triangle-free graph is a claw-free graph, but the reverse
Complement_graph
TRIANGLE GRAPH
TRIANGLE GRAPH
Surname or Lastname
English
English : unexplained; perhaps a variant of Ringle.
Surname or Lastname
English
English : metonymic occupational name for a maker of nails or pins, or nickname for a small, thin man, from Middle English tingle, a kind of very small nail (of North German origin).
Girl/Female
African, Anglo, British, Chinese, English, German, Hebrew, Swahili
To Tangle; Complication; Difficulty; Fairy Princess
Boy/Male
Spanish American Italian Latin
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Surname or Lastname
English
English : from the Old English personal name Hringwulf.German : from a short form of a Germanic personal name based on hring ‘ring’.German : metonymic occupational name for a ring maker (see Ringler).German : altered spelling of Ringel, an Old Prussian personal name.
Surname or Lastname
English (Essex, Cambridgeshire)
English (Essex, Cambridgeshire) : possibly a variant of Trendall, a topographic name for someone who lived by a well, earhwork, stone circle, or other circular feature, from Middle English trendel, trandle ‘circle’ (Old English trendel).Possibly an altered spelling of South German Tröndle, a variant of Trendle, a nickname for a tearful person, from Träne ‘tear’ + the diminutive suffix -l.
Surname or Lastname
English
English : possibly a variant of Trumble.Possibly a variant spelling of German Trindl, from a Bavarian and Swabian nickname for a slow person, or alternatively an altered spelling of Drindle, from a South German short form of the personal name Katharina (see Catherine).
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Surname or Lastname
German (also Gräff), Dutch, and Jewish (Ashkenazic)
German (also Gräff), Dutch, and Jewish (Ashkenazic) : variant of Graf.English : metonymic occupational name for a clerk or scribe, from Anglo-Norman French grafe ‘quill’, ‘pen’ (a derivative of grafer ‘to write’, Late Latin grafare, from Greek graphein).
Boy/Male
American, Anglo, Australian, British, English
From the Triangle Shaped Settlement; Lives in the Triangular Farm Stead
Surname or Lastname
English
English : occupational name from an agent derivative of Middle English tingle (see Tingle).German : occupational or status name for a medieval judge or court official, from Old High German ding ‘legal proceeding’.German : variant of Tengler.
TRIANGLE GRAPH
TRIANGLE GRAPH
Boy/Male
Sikh
Mine of nectar
Boy/Male
Tamil
Spectator
Girl/Female
Indian, Sanskrit
Older; Superior
Boy/Male
Greek
Son of Poseidon.
Girl/Female
English
Marigold (flower name). Mary's gold - refers to both the flower and the mother of Jesus.
Boy/Male
Hindu, Indian, Kannada, Sanskrit
Risen Sun
Boy/Male
Arabic, Muslim, Sindhi
Servant of the Preserver
Boy/Male
English
Son of Byrne.
Boy/Male
Hindu
Pure or holy
Boy/Male
Hindu, Indian
Fitting
TRIANGLE GRAPH
TRIANGLE GRAPH
TRIANGLE GRAPH
TRIANGLE GRAPH
TRIANGLE GRAPH
n.
A small constellation near the South Pole, containing three bright stars.
n.
Wrangle; brangle.
a.
Liable to undergo a judicial examination; properly coming under the cognizance of a court; as, a cause may be triable before one court which is not triable in another.
n.
A figure having three angles; a triangle.
n.
A kind of frame formed of three poles stuck in the ground and united at the top, to which soldiers were bound when undergoing corporal punishment, -- now disused.
n.
See Wariangle.
v. i.
To wrangle; to dispute contentiously; to squabble.
n.
An instrument of percussion, usually made of a rod of steel, bent into the form of a triangle, open at one angle, and sounded by being struck with a small metallic rod.
n.
A wrangle; a squabble; a noisy contest or dispute.
n.
A triangle having its sides and angles unequal.
n.
A figure bounded by three lines, and containing three angles.
p. pr. & vb. n.
of Strangle
n.
A small constellation situated between Aries and Andromeda.
imp. & p. p.
of Strangle
v.
A knot of threads, or other thing, united confusedly, or so interwoven as not to be easily disengaged; a snarl; as, hair or yarn in tangles; a tangle of vines and briers. Used also figuratively.
n.
A triangle having three acute angles.
v. i.
To be entangled or united confusedly; to get in a tangle.
a.
Having oblique angles; as, an oblique-angled triangle.
n.
A draughtsman's square in the form of a right-angled triangle.
n.
See Wariangle.