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For every graph G, there exists a highly irregular graph H containing G as an induced subgraph. [ 3 ] This last observation can be considered analogous to a result of Dénes Kőnig , which states that if H is a graph with greatest degree r , then there is a graph G which is r -regular and contains H as an induced subgraph.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics (graph theory) and computer science. Let S be a set of q elements and d a positive integer. The Hamming graph H(d,q) has vertex set S d, the set of ordered d-tuples of elements of S, or sequences of length d from S.
The Petersen graph is a well known non-Hamiltonian graph, but all odd graphs for are known to have a Hamiltonian cycle. [17] As the odd graphs are vertex-transitive , they are thus one of the special cases with a known positive answer to Lovász' conjecture on Hamiltonian cycles in vertex-transitive graphs.
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters. [ 1 ] In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not ...
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor.
In graph theory, a star S k is the complete bipartite graph K 1,k : a tree with one internal node and k leaves (but no internal nodes and k + 1 leaves when k ≤ 1).Alternatively, some authors define S k to be the tree of order k with maximum diameter 2; in which case a star of k > 2 has k − 1 leaves.