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According to Brooks' theorem every connected cubic graph other than the complete graph K 4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K 4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
The two edges along the cycle adjacent to any of the vertices are not written down. Let v be the vertices of the graph and describe the Hamiltonian circle along the p vertices by the edge sequence v 0 v 1, v 1 v 2, ...,v p−2 v p−1, v p−1 v 0. Halting at a vertex v i, there is one unique vertex v j at a distance d i joined by a chord with v i,
Several algorithms based on depth-first search compute strongly connected components in linear time.. Kosaraju's algorithm uses two passes of depth-first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth-first search tests vertices for having been visited already and recursively explores them if not.
This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed.. In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more ...
Mycielskian construction applied to a 5-cycle graph, producing the Grötzsch graph with 11 vertices and 20 edges, the smallest triangle-free 4-chromatic graph (Chvátal 1974). Let the n vertices of the given graph G be v 1, v 2, . . . , v n. The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices: a ...
Imagine the vertices of the graph to lie in three levels. Pick any vertex as the root, in Level 0. Then its k neighbors lie in Level 1, and all other vertices lie in Level 2. Vertices in Level 1 are directly connected to the root, hence they must have λ other neighbors in common with the root, and these common neighbors must also be in Level 1.
A cycle graph is: 2-edge colorable, if and only if it has an even number of vertices; 2-regular; 2-vertex colorable, if and only if it has an even number of vertices. More generally, a graph is bipartite if and only if it has no odd cycles (Kőnig, 1936). Connected; Eulerian; Hamiltonian; A unit distance graph; In addition:
The boxicity of a graph is the minimum dimension in which a given graph can be represented as an intersection graph of axis-parallel boxes. That is, there must exist a one-to-one correspondence between the vertices of the graph and a set of boxes, such that two boxes intersect if and only if there is an edge connecting the corresponding vertices.