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Kotzig's Conjecture
Kotzig's conjecture is an unproven assertion in graph theory which states that finite graphs with certain properties do not exist. A graph is a P_k-graph if each pair of distinct vertices is connected by ''exactly one'' path of length k. Kotzig's conjecture asserts that for k\ge 3 there are no finite P_k-graphs with two or more vertices. The conjecture was first formulated by Anton Kotzig in 1974. It has been verified for k\le 20, but remains open in the general case (as of November 2024). The conjecture is stated for k\ge 3 because P_k-graphs do exist for smaller values of k. P_1-graphs are precisely the complete graphs. The friendship theorem states that P_2-graphs are precisely the (triangular) windmill graphs (that is, finitely many triangles joined at a common vertex; also known as friendship graphs). History Kotzig's conjecture was first listed as an open problem by Bondy & Murty in 1976, attributed to Kotzig and dated to 1974. Kotzig's first own writing on the conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graceful Labeling
In graph theory, a graceful labeling of a Graph (discrete mathematics), graph with edges is a graph labeling, labeling of its Vertex (graph theory), vertices with some subset of the integers from 0 to inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and inclusive.Virginia Vassilevska Williams, Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001PostScript/ref> A graph which admits a graceful labeling is called a graceful graph. The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.. A major open problem in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC (not to be confused with Kotzig's conjecture on regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anton Kotzig
Anton Kotzig (22 October 1919 – 20 April 1991) was a Slovak–Canadian mathematician, expert in statistics, combinatorics and graph theory. A number of his mathematical contributions are named after him. These include the Ringel–Kotzig conjecture on graceful labeling of trees (with Gerhard Ringel); Kotzig's conjecture on regularly path connected graphs; Kotzig's theorem on the degrees of vertices in convex polyhedra; as well as the Kotzig transformation. Biography Kotzig was born in Kočovce, a village in Western Slovakia. He studied at the secondary grammar school in Nové Mesto nad Váhom, and began his undergraduate studies at the Charles University in Prague. After the closure of Czech universities in 1939, he moved to Bratislava where in 1943, he earned a doctoral degree (RNDr.) in Mathematical Statistics from the Comenius University. He remained in Bratislava working at the Central Bureau of Social Insurance for Slovakia as head of the Department of Mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
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 (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendship Graph
In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or -fan) is a planar, undirected graph with vertices and edges. The friendship graph can be constructed by joining copies of the cycle graph with a common vertex, which becomes a universal vertex for the graph. By construction, the friendship graph is isomorphic to the windmill graph . It is unit distance with girth 3, diameter 2 and radius 1. The graph is isomorphic to the butterfly graph. Friendship graphs are generalized by the triangular cactus graphs. Friendship theorem The friendship theorem of states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly one friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Windmill Graph
In the mathematical field of graph theory, the windmill graph is an undirected graph constructed for and by joining copies of the complete graph at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs. Properties It has vertices and edges, girth 3 (if ), radius 1 and diameter 2. It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is -edge-connected. It is trivially perfect and a block graph. Special cases By construction, the windmill graph is the friendship graph , the windmill graph is the star graph and the windmill graph is the butterfly graph. Labeling and colouring The windmill graph has chromatic number and chromatic index . Its chromatic polynomial The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Adrian Bondy
John Adrian Bondy (born 1944 in London) is a retired English mathematician, known for his work in combinatorics and graph theory. Career Bondy received his Ph.D. in graph theory from the University of Oxford in 1969. His advisor was Dominic Welsh. Between 1969 and 1994, Bondy was ''Professor of Graph Theory'' at the University of Waterloo in Canada, and then, until his retirement, at Université Lyon 1 in France. From 1976, he was managing editor, and, between 1979 and 2004, co-editor-in-chief (together with U. S. R. Murty) of Journal of Combinatorial Theory, Series B. Throughout his career, Bondy has (co-)authored over 100 publications with 51 co-authors, including the widely influential textbook ''Graph Theory with Applications'' (with U. S. R. Murty), and supervised 12 Ph.D. students. His Erdős number is 1. Bondy was dismissed from his tenured position at the University of Waterloo in 1995, after 25 years in which he had been a major contributor to the renown of the Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proofs From THE BOOK
''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book." Content ''Proofs from THE BOOK'' contains 32 sections (45 in the sixth edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory, geometry, Mathematical analysis, analysis, combinatorics and graph theory. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by . It has gone through six editions in English, and has been translated into Persian, French, German, Hungarian, Italian, Japanese, Chinese, Polish, Portuguese, Korean, Turkish, Russian, Spanish and Greek. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendship Theorem
Friendship is a relationship of mutual affection between people. It is a stronger form of interpersonal bond than an "acquaintance" or an "association", such as a classmate, neighbor, coworker, or colleague. Although there are many forms of friendship, certain features are common to many such bonds, such as choosing to be with one another, enjoying time spent together, and being able to engage in a positive and supportive role to one another. Sometimes friends are distinguished from family, as in the saying "friends and family", and sometimes from lovers (e.g., "lovers and friends"), although the line is blurred with friends with benefits. Similarly, being in the ''friend zone'' describes someone who is restricted from rising from the status of friend to that of lover (see also unrequited love). Friendship has been studied in academic fields, such as communication, sociology, social psychology, anthropology, and philosophy. Various academic theories of friendship have been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eulerian Graph
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). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this: :Given the graph in the image, is it possible to construct a path (or a cycle; i.e., a path starting and ending on the same vertex) that visits each edge exactly once? Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: :A connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
