4. Let G be a simple graph having at least one edge, and let L(G) be...
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on n vertices, that is x(Kn) n. This implies that any graph with Kn as a subgraph must have chromatic number at least n. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that...
Given an undirected connected graph so that every edge belongs to at least one simple cycle (a cycle is simple if be vertex appears more than once). Show that we can give a direction to every edge so that the graph will be strongly connected. Please write time complexity.
(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G − L has exactly two components. (b) Let G an eulerian graph. Prove that λ(G) is even.
5. Let G is a simple planar graph containing no triangles. (i) Using Euler's formula, show that G contains a vertex of degree at most 3. (ii) Use induction to deduce that G is 4-colorable-(v).
5. Let G is a simple planar graph containing no triangles. (i) Using Euler's formula, show that G contains a vertex of degree at most 3. (ii) Use induction to deduce that G is 4-colorable-(v).
Exercise 5. Let G be a graph in which every vertex has degree at least m. Prove that there is a simple path (i.e. no repeated vertices) in G of length m.
Question 1: Given an undirected connected graph so that every edge belongs to at least one simple cycle (a cycle is simple if be vertex appears more than once). Show that we can give a direction to every edge so that the graph will be strongly connected. Question 2: Given a graph G(V, E) a set I is an independent set if for every uv el, u #v, uv & E. A Matching is a collection of edges {ei} so...
a) Let G be a simple graph with degree sequence (6,6,4,4,4, 2,2). Can you guarantee that G has an Euler path? Justify your answer. b) Determine the chromatic number of the graph shown below vi V2 VS V3 VA
2) Let G ME) be an undirected Graph. A node cover of G is a subset U of the vertex set V such that every edge in E is incident to at least one vertex in U. A minimum node cover MNC) is one with the lowest number of vertices. For example {1,3,5,6is a node cover for the following graph, but 2,3,5} is a min node cover Consider the following Greedy algorithm for this problem: Algorithm NodeCover (V,E) Uempty While...
Let G be the graph obtained by removing an edge from the complete graph K By Brooks' theorem, x(G) s n-1. Give a method for (n- 1)-colouring G, and test your method by 6-colouring Ky with one edge removed