In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph. [1] Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length.
An -vertex graph is pancyclic if, for every in the range , it contains a cycle of length . [1] It is node-pancyclic or vertex-pancyclic if, for every vertex and every in the same range, it contains a cycle of length that contains . [2] Similarly, it is edge-pancyclic if, for every edge and every in the same range, it contains a cycle of length that contains . [2]
A bipartite graph cannot be pancyclic, because it does not contain any odd-length cycles, but it is said to be bipancyclic if it contains cycles of all even lengths from 4 to . [3]
A maximal outerplanar graph is a graph formed by a simple polygon in the plane by triangulating its interior. Every maximal outerplanar graph is pancyclic, as can be shown by induction. The outer face of the graph is an -vertex cycle, and removing any triangle connected to the rest of the graph by only one edge (a leaf of the tree that forms the dual graph of the triangulation) forms a maximal outerplanar graph on one fewer vertex, that by induction has cycles of all the remaining lengths. With more care in choosing which triangle to remove, the same argument shows more strongly that every maximal outerplanar graph is node-pancyclic. [4] The same holds for graphs that have a maximal outerplanar spanning subgraph, as do for instance the wheel graphs.
A maximal planar graph is a planar graph in which all faces, even the outer face, are triangles. A maximal planar graph is node-pancyclic if and only if it has a Hamiltonian cycle: [5] if it is not Hamiltonian, it is certainly not pancyclic, and if it is Hamiltonian, then the interior of the Hamiltonian cycle forms a maximal outerplanar graph on the same nodes, to which the previous argument for maximal outerplanar graphs can be applied. [6] For instance, the illustration shows the pancyclicity of the graph of an octahedron, a Hamiltonian maximal planar graph with six vertices. More strongly, by the same argument, if a maximal planar graph has a cycle of length , it has cycles of all smaller lengths. [7]
Halin graphs are the planar graphs formed from a planar drawing of a tree that has no degree-two vertices, by adding a cycle to the drawing that connects all the leaves of the tree. Halin graphs are not necessarily pancyclic, but they are almost pancyclic in the sense that there is at most one missing cycle length. The length of the missing cycle is necessarily even. If none of the interior vertices of a Halin graph has degree three, then it is necessarily pancyclic. [8]
Bondy (1971) observed that many classical conditions for the existence of a Hamiltonian cycle were also sufficient conditions for a graph to be pancyclic, and on this basis conjectured that every 4-connected planar graph is pancyclic. However, Malkevitch (1971) found a family of counterexamples.
A tournament is a directed graph with one directed edge between each pair of vertices. Intuitively, a tournament can be used to model a round-robin sports competition, by drawing an edge from the winner to the loser of each game in the competition. A tournament is called strongly connected or strong if and only if it cannot be partitioned into two nonempty subsets and of losers and winners, such that every competitor in beats every competitor in . [9] Every strong tournament is pancyclic [10] and node-pancyclic. [11] If a tournament is regular (each competitor has the same number of wins and losses as each other competitor) then it is also edge-pancyclic; [12] however, a strong tournament with four vertices cannot be edge-pancyclic.
Mantel's theorem states that any -vertex undirected graph with at least edges, and no multiple edges or self-loops, either contains a triangle or it is the complete bipartite graph . This theorem can be strengthened: any undirected Hamiltonian graph with at least edges is either pancyclic or . [1]
There exist -vertex Hamiltonian directed graphs with edges that are not pancyclic, but every Hamiltonian directed graph with at least edges is pancyclic. Additionally, every -vertex strongly connected directed graph in which each vertex has degree at least (counting incoming and outgoing edges together) is either pancyclic or it is a complete bipartite directed graph. [13]
For any graph , its kth power is defined as the graph on the same vertex set that has an edge between every two vertices whose distance in is at most . If is 2-vertex-connected, then by Fleischner's theorem its square is Hamiltonian; this can be strengthened to show that it is necessarily vertex-pancyclic. [14] More strongly, whenever is Hamiltonian, it is also pancyclic. [15]
It is NP-complete to test whether a graph is pancyclic, even for the special case of 3-connected cubic graphs, and it is also NP-complete to test whether a graph is node-pancyclic, even for the special case of polyhedral graphs. [16] It is also NP-complete to test whether the square of a graph is Hamiltonian, and therefore whether it is pancyclic. [17]
Pancyclicity was first investigated in the context of tournaments by Harary & Moser (1966), Moon (1966), and Alspach (1967). The concept of pancyclicity was named and extended to undirected graphs by Bondy (1971).
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 only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details.
Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to Kelly and Ulam.
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs or triangulated graphs: a chordal completion of a graph is typically called a triangulation of that graph.
In the mathematical discipline of graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can also be defined as the 1-skeleton of an (n – 1)-gonal pyramid.
In graph theory, the hypercube graphQn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Qn has 2n vertices, 2n – 1n edges, and is a regular graph with n edges touching each vertex.
In graph theory, a cactus is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or in which every block is an edge or a cycle.
In the mathematical field of graph theory, a graph G is said to be hypohamiltonian if G itself does not have a Hamiltonian cycle but every graph formed by removing a single vertex from G is Hamiltonian.
In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the plane so none of its edges cross, and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.
In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian. The result has been widely used to prove that certain planar graphs constructed to have additional properties are not Hamiltonian; for instance it can prove non-Hamiltonicity of some counterexamples to Tait's conjecture that cubic polyhedral graphs are Hamiltonian.
In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs. In the undirected case a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that each factor is connected.
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs.
In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph, and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. It is named after British astronomer Alexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs.
In graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path.
In graph theory, a mathematical discipline, coloring refers to an assignment of colours or labels to vertices, edges and faces of a graph. Defective coloring is a variant of proper vertex coloring. In a proper vertex coloring, the vertices are coloured such that no adjacent vertices have the same colour. In defective coloring, on the other hand, vertices are allowed to have neighbours of the same colour to a certain extent.
In graph theory, a branch of mathematics, Fleischner's theorem gives a sufficient condition for a graph to contain a Hamiltonian cycle. It states that, if is a 2-vertex-connected graph, then the square of is Hamiltonian. It is named after Herbert Fleischner, who published its proof in 1974.
In graph theory and graph drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph.
In graph theory, the act of coloring generally implies the assignment of labels to vertices, edges or faces in a graph. The incidence coloring is a special graph labeling where each incidence of an edge with a vertex is assigned a color under certain constraints.
