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mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte. It has chromatic number
Tutte_graph
British-Canadian codebreaker and mathematician (1917–2002)
fields of graph theory and matroid theory. Tutte's research in the field of graph theory proved to be of remarkable importance. At a time when graph theory
W._T._Tutte
Algebraic encoding of graph connectivity
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays
Tutte_polynomial
3-regular graph with 30 vertices and 45 edges
mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45
Tutte–Coxeter_graph
Characterization of the size of a maximum matching in a graph
discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte's theorem
Tutte–Berge_formula
Planar graph drawn by relaxing springs
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free
Tutte_embedding
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which
Tutte_matrix
Methodic assignment of colors to elements of a graph
which was generalised to the Tutte polynomial by W. T. Tutte, both of which are important invariants in algebraic graph theory. Kempe had already drawn
Graph_coloring
Area of discrete mathematics
Algebraic graph theory also studies the algebraic invariants, chromatic polynomial, Tutte polynomial of a graph, and knot invariant. A graph invariant
Graph_theory
Concept in graph theory
In graph theory, a Tutte path is a path P {\displaystyle P} within a graph G {\displaystyle G} such that every connected component that remains after
Tutte_path
Graph made from vertices and edges of a convex polyhedron
Given such a graph, a representation of it as a subdivision of a convex polygon into smaller convex polygons may be found using the Tutte embedding. Tait
Polyhedral_graph
Disproven graph theory
cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by P. G. Tait (1884) and disproved by W. T. Tutte (1946)
Tait's_conjecture
Regular graph with fewest possible nodes for its girth
Heawood graph, 14 vertices (3,7)-cage: the McGee graph, 24 vertices (3,8)-cage: the Tutte–Coxeter graph, 30 vertices (3,10)-cage: the Balaban 10-cage, 70
Cage_(graph_theory)
Sylvester graph Tutte's fragment Tutte graph Young–Fibonacci graph Wagner graph Wells graph Wiener–Araya graph Windmill graph The strongly regular graph on v
List_of_graphs
Graph with all vertices of degree 3
Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified
Cubic_graph
On Hamiltonian cycles in planar graphs
In graph theory, a theorem of W. T. Tutte states that every 4-vertex-connected planar graph has a Hamiltonian cycle. It strengthens an earlier theorem
Tutte's theorem on Hamiltonian cycles
Tutte's_theorem_on_Hamiltonian_cycles
Characterization of graphs with perfect matchings
mathematical discipline of graph theory, the Tutte theorem, named after William Thomas Tutte, is a characterization of finite undirected graphs with perfect matchings
Tutte's theorem on perfect matchings
Tutte's_theorem_on_perfect_matchings
Bipartite non-Hamiltonian polyhedral graph
polyhedral 3-regular graph must be Hamiltonian; this was disproved when W. T. Tutte provided a counterexample, the Tutte graph, which is much larger
Herschel_graph
Planar bipartite graph with 25 vertices and 31 edges
In the mathematical field of graph theory, the Walther graph, also called the Tutte fragment, is a planar bipartite graph with 25 vertices and 31 edges
Walther_graph
Non-Hamiltonian simple polyhedron
theorem. The Barnette–Bosák–Lederberg graph has a similar construction to the Tutte graph but is composed of two Tutte fragments, connected through a pentagonal
Barnette–Bosák–Lederberg graph
Barnette–Bosák–Lederberg_graph
Cubic graph with 10 vertices and 15 edges
bridgeless graph has a cycle-continuous mapping to the Petersen graph. More unsolved problems in mathematics In the mathematical field of graph theory, the
Petersen_graph
Topics referred to by the same term
4-vertex-connected planar graphs Tutte's theorem on perfect matchings, a characterization of the graphs having perfect matchings Tutte's spring theorem, on the
Tutte's_theorem
Graph that can be embedded in the plane
of certain Schrödinger operators defined by the graph. The Hanani–Tutte theorem states that a graph is planar if and only if it has a drawing in which
Planar_graph
On parity of crossings in graph drawings
In topological graph theory, the Hanani–Tutte theorem is a result on the parity of edge crossings in a graph drawing. It states that every drawing in
Hanani–Tutte_theorem
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
78-graph is 3 and the queue numbers 2. The Ellingham-Horton 54-graph is 1-planar. The first counterexample to the Tutte conjecture was the Horton graph,
Ellingham–Horton_graph
Physical simulation to visualize graphs
force-directed graph-drawing", Journal of Graph Algorithms and Applications, 7 (3): 253–285, doi:10.7155/jgaa.00070, MR 2112231 Tutte, W. T. (1963), "How
Force-directed_graph_drawing
Cycles in a graph that cover each edge twice
by W. T. Tutte, Itai and Rodeh, George Szekeres and Paul Seymour and known as the cycle double cover conjecture, whether every bridgeless graph has a cycle
Cycle_double_cover
Geometric graph with unit edge lengths
In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting
Unit_distance_graph
Regular graph with girth more than twice its diameter
Does a Moore graph with girth 5 and degree 57 exist? More unsolved problems in mathematics In graph theory, a Moore graph is a regular graph whose girth
Moore_graph
Function in algebraic graph theory
\left\{(0,P(G,0)),(1,P(G,1)),\ldots ,(n,P(G,n))\right\}.} Tutte’s curiosity about which other graph invariants satisfied such recurrences led him to discover
Chromatic_polynomial
3-regular graph with no 3-edge-coloring
as the problems they mention, W. T. Tutte's snark conjecture concerns the existence of Petersen graphs as graph minors of snarks; its proof has been
Snark_(graph_theory)
Trail in a graph that visits each edge once
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices)
Eulerian_path
Length of a shortest cycle contained in the graph
graph and the Harries–Wong graph. The Petersen graph has a girth of 5 The Heawood graph has a girth of 6 The McGee graph has a girth of 7 The Tutte–Coxeter
Girth_(graph_theory)
The Tutte polynomial of the Chvátal graph has been computed by Björklund et al. (2008). The independence number of this graph is 4. The graph is 1-planar
Chvátal_graph
Graph cycle which does not separate remaining elements
because Tutte called cycles "polygons") were first studied by Tutte (1963), and play important roles in the characterization of planar graphs and in generating
Peripheral_cycle
Maximal subgraph whose vertices can reach each other
play a key role in Tutte's theorem on perfect matchings characterizing finite graphs that have perfect matchings and the associated Tutte–Berge formula for
Component_(graph_theory)
Property of graphs that depends only on abstract structure
polynomial, such as the Tutte polynomial of a graph. Easily computable graph invariants are instrumental for fast recognition of graph isomorphism, or rather
Graph_property
Formula used in graph theory
Tatyana van Aardenne-Ehrenfest, Cedric Smith and W. T. Tutte. Let G = (V, E) be a directed graph. An Eulerian circuit is a directed closed trail that visits
BEST_theorem
Abstraction of linear independence of vectors
invariant is an evaluation of the Tutte polynomial. The Tutte polynomial T G {\displaystyle T_{G}} of a graph is the Tutte polynomial T M ( G ) {\displaystyle
Matroid
Graph where any two nodes of equal distance are isomorphic
distance-transitive is the Shrikhande graph, with 16 vertices and degree 6. The only graph of this type with degree three is the 126-vertex Tutte 12-cage. Complete lists
Distance-transitive_graph
Path in a graph that visits each vertex exactly once
(1931), "A theorem on graphs", Annals of Mathematics, Second Series, 32 (2): 378–390, doi:10.2307/1968197, JSTOR 1968197, MR 1503003 Tutte, W. T. (1956), "A
Hamiltonian_path
Graph property
Cubical graph, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the Dodecahedral graph, the Desargues graph, Tutte 12-cage
Distance-regular_graph
Concept in graph theory and network analysis
In graph theory and network analysis, group centrality generalizes the concept of centrality to sets of nodes in a network. Introduced by Everett and
Group_centrality
Graph without triples of adjacent vertices
triangle free graphs with arbitrarily high chromatic number is due to Tutte (writing as Blanche Descartes). This construction started from the graph with a single
Triangle-free_graph
Index of articles associated with the same name
a graph. The reliability polynomial, a polynomial that describes the probability of remaining connected after independent edge failures The Tutte polynomial
Graph_polynomial
of graph theory, the Tutte 12-cage or Benson graph is a 3-regular graph with 126 vertices and 189 edges. It is named after W. T. Tutte. The Tutte 12-cage
Tutte_12-cage
Set of edges without common vertices
for bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and Tutte's theorem on perfect
Matching_(graph_theory)
Directed graph where every node has exactly one path to it from the root
Springer Science & Business Media. p. 28. ISBN 978-3-642-24488-9. Tutte, W.T. (2001), Graph Theory, Cambridge University Press, pp. 126–127, ISBN 978-0-521-79489-3
Arborescence_(graph_theory)
Graph representing incident points and lines
of 27 points and the 27 orthogonal lines through them. The Tutte eight-cage is the Levi graph of the Cremona–Richmond configuration. It is also known as
Levi_graph
whether a graph is q-tough is co-NP-complete (Bauer, Hakimi & Schmeichel 1990). Strength of a graph, an analogous concept for edge deletions Tutte–Berge formula
Graph_toughness
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
Theorem on edge-disjoint spanning trees
independently by Tutte and Nash-Williams, both in 1961. In 2012, Kaiser gave a short elementary proof. For this article, we say that such a graph has arboricity t
Nash-Williams_theorem
Tree which includes all vertices of a graph
of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may
Spanning_tree
Graph with 24 vertices and 36 edges
published the result in 1960. Then, the McGee graph was proven the unique (3,7)-cage by Tutte in 1966. The McGee graph requires at least eight crossings in any
McGee_graph
Subgraph with contracted edges
In graph theory, an undirected graph H is called a minor of the undirected graph G if H can be formed from G by deleting edges and vertices and by contracting
Graph_minor
Edge-face adjacencies in another graph
graph theory, the medial graph of plane graph G is another graph M(G) that represents the adjacencies between edges in the faces of G. Medial graphs were
Medial_graph
Branch of mathematics
branch of algebraic graph theory concerns algebraic properties of invariants of graphs, and especially the chromatic polynomial, the Tutte polynomial and knot
Algebraic_graph_theory
Concept in graph theory
form a group as the sum of two flows on one edge may add to 0. (Tutte 1950) A graph G has an M-flow if and only if it has a |M|-flow. As a consequence
Nowhere-zero_flow
Problem of finding a cycle through all vertices of a graph
graphs can be carried out in linear time by computing a so-called Tutte path. Tutte proved this result by showing that every 2-connected planar graph
Hamiltonian_path_problem
On the number of spanning trees in a graph
and cocycles in graphs", SIAM Journal on Applied Mathematics, 30 (1): 143–148, doi:10.1137/0130017, MR 0392635. Tutte, W. T. (2001), Graph Theory, Cambridge
Kirchhoff's_theorem
Graph which remains connected when k or fewer nodes removed
{\displaystyle k} -linked. k-edge-connected graph Connectivity (graph theory) Menger's theorem Structural cohesion Tutte embedding Vertex separator Schrijver
Vertex_connectivity
Number of planar subgraphs to cover a graph
within an approximation ratio of 3 in polynomial time. Tutte, W. T. (1963), "The thickness of a graph", Indag. Math., 66: 567–577, doi:10.1016/S1385-7258(63)50055-9
Thickness_(graph_theory)
Directed graph isomorphic to its own transpose graph
digraphs by Tutte (1967), later as the double covering graphs of polar graphs by Zelinka (1976b), and still later as the double covering graphs of bidirected
Skew-symmetric_graph
Type of graph invariant
Tutte–Grothendieck (TG) invariant is a type of graph invariant that satisfies a generalized deletion–contraction formula. Any evaluation of the Tutte
Tutte–Grothendieck_invariant
Fewest edge crossings in drawing of a graph
graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is
Crossing number (graph theory)
Crossing_number_(graph_theory)
its maximum degree? Tutte's conjectures: every bridgeless graph has a nowhere-zero 5-flow every Petersen-minor-free bridgeless graph has a nowhere-zero
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
a counterexample to the Tutte conjecture that every cubic 3-connected bipartite graph is Hamiltonian. After the Horton graph, a number of smaller counterexamples
Horton_graph
Unsolved problem in graph theory
these known counterexamples is bipartite. Tutte himself conjectured that every cubic 3-connected bipartite graph is Hamiltonian, but this was shown to be
Barnette's_conjecture
Graph-theoretic description of polyhedra
3-connected graph is the graph of a convex polyhedron. There are three standard approaches for this part: proofs by induction, lifting two-dimensional Tutte embeddings
Steinitz's_theorem
2018 mathematics book by Marcus Schaefer
and the Hanani–Tutte theorem on the parity of crossings. The second chapter concerns other special classes of graphs including graph products (especially
Crossing_Numbers_of_Graphs
Formula in graph theory
} Here G is a graph, f is a function on graphs, e is any edge of G, G \ e denotes edge deletion, and G / e denotes contraction. Tutte refers to such
Deletion–contraction_formula
1998 book by Fan Chung
on Graphs", The Mathematical Gazette, 85 (503): 375–377, doi:10.2307/3622075, JSTOR 3622075 Tutte, W. T. (September 2000), "Review of Erdős on Graphs",
Erdős_on_Graphs
Balanced complete multipartite graph
The Turán graph, denoted by T ( n , r ) {\displaystyle T(n,r)} , is a complete multipartite graph; it is formed by partitioning a set of n {\displaystyle
Turán_graph
Branch of discrete mathematics
cycles) to algebraic representations (e.g., given a graph G and two numbers x and y, does the Tutte polynomial TG(x,y) have a combinatorial interpretation
Combinatorics
Cycle graph with all opposite nodes linked
However, the 10-vertex cubic graph with the most spanning trees is the Petersen graph, which is not a Möbius ladder. The Tutte polynomials of the Möbius
Möbius_ladder
On Hamiltonian cycles in planar graphs
than one cyclic component. The 46-vertex Tutte graph, and the smaller cubic non-Hamiltonian polyhedral graphs derived from it, have cyclic edge connectivity
Grinberg's_theorem
Graph with only one possible coloring
3-edge-colorable contains a triangle, but W. T. Tutte (1976) observed that the generalized Petersen graph G(9,2) is non-planar, triangle-free, and uniquely
Uniquely_colorable_graph
4-regular undirected graph with 70 vertices and 140 edges
every 4-regular 4-vertex-connected graph is Hamiltonian. However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian. The characteristic
Meredith_graph
On tangency patterns of circles
these drawings resolves a problem posed by W. T. Tutte in 1963. A pointed drawing of a given planar graph, in which the edges are drawn as smooth curves
Circle_packing_theorem
On composition of paths in matroids
In mathematics, Tutte's homotopy theorem, introduced by Tutte (1958), generalises the concept of "path" from graphs to matroids, and states roughly that
Tutte_homotopy_theorem
7-regular undirected graph with 50 nodes and 175 edges
Singleton graph also contains the odd graph O(4), the Coxeter graph, and the Tutte-Coxeter graph as subgraphs. Take any edge of the Hoffman-Singleton graph, and
Hoffman–Singleton_graph
Mathematical puzzle of avoiding crossings
for the Promotion of Mechanism and Machine Science Tutte, W. T. (1947), "A family of cubical graphs", Proceedings of the Cambridge Philosophical Society
Three_utilities_problem
Integer associated with a graph
in 1965 by Paul Erdős, Frank Harary and William Tutte. It generalises the concept of unit distance graph to more than 2 dimensions. In the worst case, every
Dimension_(graph_theory)
Characterization of planar graphs by matroids
"Non-separable and planar graphs", Transactions of the American Mathematical Society, 34 (2): 339–362, doi:10.1090/S0002-9947-1932-1501641-2. Tutte, W. T. (1965)
Whitney's_planarity_criterion
Graph that is edge-transitive and regular but not vertex-transitive
graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is
Semi-symmetric_graph
Canadian geometer (1907–2003)
My Graph[P83] showed a 3-regular graph with 28 vertices and 42 edges. A graph with 30 vertices and 45 edges was discovered by William Thomas Tutte, and
H.S.M._Coxeter
Graph able to be embedded on a torus
the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices and edges can be
Toroidal_graph
In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and
Topological_graph
machine; namesake of Tutte's theorem on perfect matchings, Tutte matrix, Tutte graph, Tutte–Coxeter graph, Tutte 12-cage and Tutte fragment Abraham Robinson
List of University of Toronto faculty
List_of_University_of_Toronto_faculty
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices ( u 1 , v 1 )
Symmetric_graph
self-complementary graph, a block graph, a split graph, an interval graph, a claw-free graph, a 1-vertex-connected graph and a 1-edge-connected graph. A graph is bull-free
Bull_graph
Unproven generalization of the four-color theorem
theorem, that every cubic graph requiring four colors in any edge coloring has the Petersen graph as a minor, conjectured by W. T. Tutte and announced to be
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
Matroid with graph forests as independent sets
MR 0597159. Gerards, A. M. H. (1995), "On Tutte's characterization of graphic matroids—a graphic proof", Journal of Graph Theory, 20 (3): 351–359, doi:10.1002/jgt
Graphic_matroid
Mathematical structure
to the structural rigidity of graphs and their ability to be decomposed into edge-disjoint spanning trees via the Tutte and Nash-Williams theorem. There
Sparsity_matroid
Eppstein. "Partition a graph into node-disjoint cycles". Tutte, W. T. (1954), "A short proof of the factor theorem for finite graphs" (PDF), Canadian Journal
Vertex_cycle_cover
Planar graphs have straight drawings
a maximal planar graph into three trees known as a Schnyder wood. Tutte's spring theorem states that every 3-connected planar graph can be drawn on a
Fáry's_theorem
On Hamiltonian cycles in toroidal graphs
states that every 4-vertex-connected toroidal graph has a Hamiltonian cycle. It is a generalization of Tutte's theorem on Hamiltonian cycles, according to
Grünbaum–Nash-Williams conjecture
Grünbaum–Nash-Williams_conjecture
Element of graph theory
orientations of planar graphs extends in this form to nonplanar graphs as well: the Tutte polynomial of the dual graph of a planar graph is obtained by swapping
Acyclic_orientation
Matching which covers every node of the graph
of bipartite graphs which have a perfect matching. Tutte's theorem on perfect matchings provides a characterization for arbitrary graphs. A perfect matching
Perfect_matching
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