Let G = (V, E) be a graph with |V| = n, and let k be an integer, where 1 lesstha
ID: 3862550 • Letter: L
Question
Let G = (V, E) be a graph with |V| = n, and let k be an integer, where 1 lessthanorequalto k lessthanorequalto n. Prove the following theorem: "Suppose the vertices in V can be ordered upsilon_1, upsilon_2, ..., upsilon_3 in such a wage that each vertex upsilon_1 has at most k neighbors among the preceding vertices upsilon_1, ..., upsilon_i - 1. Then G can be colored with at most k + 1 colors." For example, the figure below shows a graph and an ordering of its vertices where each vertex has at most 3 neighbors that precede it in this ordering. So this theorem claims that this graph can be colored with 4 colors. (b) Prove that the theorem in part (a) above implies t lie following statement: "If all vertices of a graph G have degree at most D then G can be colored with at most D + 1 colors". (You will cover a different proof of this theorem in the discussion. Here you only need to show how it can be derived from part (a).)Explanation / Answer
A graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices. A graph that is not connected is disconnected.
A graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices. A graph that is not connected is disconnected. A graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints.
A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected.
n an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are calleddisconnected. If the two vertices are additionally connected by a path of length 1, i.e. by a single edge, the vertices are called adjacent. Agraph is said to be connected if every pair of vertices in the graph is connected.
A connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected component, as does each edge.
A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is connected if it contains a directed path from u to v or a directed path from v to u for every pair of vertices u, v. It is strongly connected, diconnected, or simply strong if it contains a directed path from u to v and a directed path from v to u for every pair of verticesu, v. The strong components are the maximal strongly connected subgraphs.
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