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Gary Chartrand was born in 1936. He was raised in Sault Ste. Marie, Michigan and attended J. W. Sexton High School located in Lansing, Michigan.As an undergraduate student, he initially majored in chemical engineering, but switched to mathematics in his junior year, in which he also became a member of the honorary mathematics society Pi Mu Epsilon.
In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1) / 2 . The edges of an undirected simple graph permitting loops G {\displaystyle G} induce a symmetric homogeneous relation ∼ {\displaystyle \sim } on the vertices of G {\displaystyle G} that is called ...
A graceful labeling. Vertex labels are in black, edge labels in red.. In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers from 0 to m 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 m ...
For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. ch(G) ≥ χ(G).A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than the maximum degree Δ of the graph. At least Δ colors are always necessary, so the undirected graphs may be partitioned into two classes: "class one" graphs for which Δ colors suffice, and ...
The cube of every connected graph necessarily contains a Hamiltonian cycle. [10] It is not necessarily the case that the square of a connected graph is Hamiltonian, and it is NP-complete to determine whether the square is Hamiltonian. [11] Nevertheless, by Fleischner's theorem, the square of a 2-vertex-connected graph is always Hamiltonian. [12]
For a plane graph G, twice the evaluation of the Tutte polynomial at the point (3,3) equals the sum over weighted Eulerian orientations in the medial graph of G, where the weight of an orientation is 2 to the number of saddle vertices of the orientation (that is, the number of vertices with incident edges cyclically ordered "in, out, in out"). [5]
Another equivalent definition of a Moore graph G is that it has girth g = 2k + 1 and precisely n / g (m – n + 1) cycles of length g, where n and m are, respectively, the numbers of vertices and edges of G. They are in fact extremal with respect to the number of cycles whose length is the girth of the graph. [1]