Quartic graph

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In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph. [1]

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Graph theory Area of discrete mathematics

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices which are connected by edges. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

Graph (discrete mathematics) Mathematical structure consisting of vertices and edges connecting some pairs of vertices

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

Contents

Examples

The Chvatal graph Chvatal Lombardi.svg
The Chvátal graph

Several well-known graphs are quartic. They include:

Complete graph simple undirected graph in which every pair of distinct vertices is connected by a unique edge

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 edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges.

Chvátal graph a triangle-free 4-regular 4-chromatic graph with 12 vertices and 24 edges

In the mathematical field of graph theory, the Chvátal graph is an undirected graph with 12 vertices and 24 edges, discovered by Václav Chvátal (1970).

Graph coloring assignment of labels traditionally called "colors" to elements of a graph

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

Every medial graph is a quartic plane graph, and every quartic plane graph is the medial graph of a pair of dual plane graphs or multigraphs. [5] Knot diagrams and link diagrams are also quartic plane multigraphs, in which the vertices represent the crossings of the diagram and are marked with additional information concerning which of the two branches of the knot crosses the other branch at that point. [6]

Medial graph

In the mathematical discipline of 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 introduced in 1922 by Ernst Steinitz to study combinatorial properties of convex polyhedra, although the inverse construction was already used by Peter Tait in 1877 in his foundational study of knots and links.

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 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.

Multigraph graph which is permitted to have multiple edges

In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges, that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge.

Properties

Because the degree of every vertex in a quartic graph is even, every connected quartic graph has an Euler tour. And as with regular bipartite graphs more generally, every bipartite quartic graph has a perfect matching. In this case, a much simpler and faster algorithm for finding such a matching is possible than for irregular graphs: by selecting every other edge of an Euler tour, one may find a 2-factor, which in this case must be a collection of cycles, each of even length, with each vertex of the graph appearing in exactly one cycle. By selecting every other edge again in these cycles, one obtains a perfect matching in linear time. The same method can also be used to color the edges of the graph with four colors in linear time. [7]

Degree (graph theory) number of edges incident to a given vertex in a node-link graph

In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex is denoted or . The maximum degree of a graph G, denoted by Δ(G), and the minimum degree of a graph, denoted by δ(G), are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, all degrees are the same, and so we can speak of the degree of the graph. A complete graph is a special kind of regular graph where all vertices,p ,have the maximum degree, p-1. A complete graph is denoted with the form Kp where p is the number of vertices in the graph.

Bipartite graph graph of two sets in which every vertex in one set is connected to at least one in the other

In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and such that every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

Algorithm An unambiguous specification of how to solve a class of problems

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, and other tasks.

Quartic graphs have an even number of Hamiltonian decompositions. [8]

Hamiltonian decomposition

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. For such a decomposition to exist in an undirected graph, it must be connected and regular of even degree. A directed graph with such a decomposition must be strongly connected and all vertices must have the same in-degree and out-degree as each other, but this degree does not need to be even.

Open problems

It is an open conjecture whether all quartic Hamiltonian graphs have an even number of Hamiltonian circuits, or have more than one Hamiltonian circuit. The answer is known to be false for quartic multigraphs. [9]

See also

Related Research Articles

Hamiltonian path path in a graph that visits each vertex exactly once

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 Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete.

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 of G. The name line graph comes from a paper by Harary & Norman (1960) although both Whitney (1932) and Krausz (1943) used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the ϑ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph.

Edge coloring an assignment of colors to the edges of a graph so that no two edges that share an endpoint have the same color as each other

In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

Cubic graph node-link graphs in which every vertex is incident to exactly three edges

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.

Heawood graph undirected graph with 14 vertices, 21 edges, and girth 6

In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.

Semi-symmetric graph

In the mathematical field of 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 if each vertex has the same number of incident edges, and there is a symmetry taking any of the graph's edges to any other of its edges, but there is some pair of vertices such that no symmetry maps the first into the second.

Gray graph undirected bipartite graph with 54 vertices and 81 edges

In the mathematical field of graph theory, the Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive.

Ores theorem theorem that a graph in which every two nonadjacent vertices have high degree sum must have a Hamiltonian cycle

Ore's theorem is a result in graph theory proved in 1960 by Norwegian mathematician Øystein Ore. It gives a sufficient condition for a graph to be Hamiltonian, essentially stating that a graph with sufficiently many edges must contain a Hamilton cycle. Specifically, the theorem considers the sum of the degrees of pairs of non-adjacent vertices: if every such pair has a sum that at least equals the total number of vertices in the graph, then the graph is Hamiltonian.

Hypercube graph

In graph theory, the hypercube graphQn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cubical 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.

Factor-critical graph

In graph theory, a mathematical discipline, a factor-critical graph is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching.

Václav Chvátal Czech-Canadian mathematician

Václav (Vašek) Chvátal (Czech: [ˈvaːtslaf ˈxvaːtal] is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada. He has published extensively on topics in graph theory, combinatorics, and combinatorial optimization.

Hypohamiltonian graph graph G is said to be hypohamiltonian if G does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from G is Hamiltonian

In the mathematical field of graph theory, a graph G is said to be hypohamiltonian if G does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from G is Hamiltonian.

Generalized Petersen graph

In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins.

Folkman graph graph with 20 vertices and 40 edges, the smallest semi-symmetric graph

In the mathematical field of graph theory, the Folkman graph, named after Jon Folkman, is a bipartite 4-regular graph with 20 vertices and 40 edges.

Ellingham–Horton graph

In the mathematical field of graph theory, the Ellingham–Horton graphs are two 3-regular graphs on 54 and 78 vertices: the Ellingham–Horton 54-graph and the Ellingham–Horton 78-graph. They are named after Joseph D. Horton and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian. The book thickness of the Ellingham-Horton 54-graph and the Ellingham-Horton 78-graph is 3 and the queue numbers 2.

Herschel graph bipartite undirected graph

In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel, who wrote an early paper concerning William Rowan Hamilton's icosian game: the Herschel graph describes the smallest convex polyhedron for which this game has no solution. However, Herschel's paper described solutions for the Icosian game only on the graphs of the regular tetrahedron and regular icosahedron; it did not describe the Herschel graph.

Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.

Pancyclic graph

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. Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length.

Fleischners theorem

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 G is a 2-vertex-connected graph, then the square of G is Hamiltonian. it is named after Herbert Fleischner, who published its proof in 1974.

References

  1. ↑ Toida, S. (1974), "Construction of quartic graphs", Journal of Combinatorial Theory , Series B, 16: 124–133, doi:10.1016/0095-8956(74)90054-9, MR   0347693 .
  2. ↑ Chvátal, V. (1970), "The smallest triangle-free 4-chromatic 4-regular graph", Journal of Combinatorial Theory , 9 (1): 93–94, doi:10.1016/S0021-9800(70)80057-6 .
  3. ↑ Folkman, Jon (1967), "Regular line-symmetric graphs", Journal of Combinatorial Theory , 3: 215–232, doi:10.1016/s0021-9800(67)80069-3, MR   0224498 .
  4. ↑ Meredith, G. H. J. (1973), "Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs", Journal of Combinatorial Theory , Series B, 14: 55–60, doi:10.1016/s0095-8956(73)80006-1, MR   0311503 .
  5. ↑ Bondy, J. A.; Häggkvist, R. (1981), "Edge-disjoint Hamilton cycles in 4-regular planar graphs", Aequationes Mathematicae , 22 (1): 42–45, doi:10.1007/BF02190157, MR   0623315 .
  6. ↑ Welsh, Dominic J. A. (1993), "The complexity of knots", Quo vadis, graph theory?, Annals of Discrete Mathematics, 55, Amsterdam: North-Holland, pp. 159–171, doi:10.1016/S0167-5060(08)70385-6, MR   1217989 .
  7. ↑ Gabow, Harold N. (1976), "Using Euler partitions to edge color bipartite multigraphs", International Journal of Computer and Information Sciences, 5 (4): 345–355, doi:10.1007/bf00998632, MR   0422081 .
  8. ↑ Thomason, A. G. (1978), "Hamiltonian cycles and uniquely edge colourable graphs", Annals of Discrete Mathematics, 3: 259–268, doi:10.1016/s0167-5060(08)70511-9, MR   0499124 .
  9. ↑ Fleischner, Herbert (1994), "Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs", Journal of Graph Theory, 18 (5): 449–459, doi:10.1002/jgt.3190180503, MR   1283310 .
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