G ( p, d ) is a cubic (3-valent) graph consisting of a p -gon and a ( p/d )-gon (a starpolygon) with corresponding vertices joined (the notation admits anomalous cases, when d =1 or ( d, p )>1), and with a high degree of symmetry. It is shown here that the seven ...
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, nodes, or points which are connected by edges, arcs, or lines. — Wikipedia D3 Graph Theory is a project...
1. What is the sum of the degrees of the vertices of a tree with n vertices? 2. Prove that a tree with at least two vertices has at least two vertices of degree 1. How many vertices does a regular graph of degree 4 with 10 edges have?
For a graph G, let [Math Processing Error]f2(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of [Math Processing Error]f2(G) over 3-regular n-vertex simple graphs G. To do this, we prove that every 3-regular multigraph with exactly ...
Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Extending a perfe...
Note on $4$-coloring $6$-regular triangulations on the torus In 1973, Altshuler characterized the 6 6 6 -regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the r imes s r imes s toroidal grid where the vertices in the first and ...
Then P(G)has at most p+1 affinely independent vertices. By Caratheodorys theorem,if wP(G),then w belongs to a convex hull of some affinely independent vertices of this polytope(see Theorem 5.1 of [9]). So w belongs to a q-vertex-polytope Q of P(G)with qp + 1. The following ...
graph,t,(G)≥6,G isa n-cycle in G.Ifthereexistsatmostonesetof3verticesof degree3,IE(c,1)n ER(G)I≥1. Forremovable edges outsidea spanning treein3-connected graphs,wegive oneresultbasedontheresearchin 3-regulargraph. Asfollows: ...
You are given a bipartite graph with nn vertices and mm edges. Each edge has a cost cece. You want to find the minimum cost perfect matching: each vertex on the left must be matched to exactly one vertex in the right and vice versa. The sum of the costs of the edges that are in ...
D3 Graph Theory 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, nodes, or points which are connected by edges, arcs, or lines. -- Wikipedia ...