Suppose that x is a tree such that for every vertex v of x, (deg(v))%3=1.
Prove that x cannot have 25 vertices.
Suppose that x is a tree such that for every vertex v of x, (deg(v))%3=1. Prove...
Please answer the questions in detail with all working and maths Suppose that T is a tree such that for every vertex v of T. (deg())%3 = 1. Prove that I cannot have 25 vertices.
1. Suppose the address of vertex v in the ordered rooted tree T is 4.5.4.6. At what level is v? What is the address of the parent of v? What is the least number of siblings v can have? What is the smallest possible number of vertices in T? If v has two children, what are their addresses? 2. Suppose the address of vertex v in the ordered rooted tree T is 4.3.5.3.4. At what level is v? What is...
Long paths we show that for every n ≥ 3 if deg(v) ≥ n/2 for every v ∈ V then the graph contains a simple cycle (no vertex appears twice) that contains all vertices. Such a path is called an Hamiltonian path. From now on we assume that deg(v) ≥ n/2 for every v. 1. Show that the graph is connected (namely the distance between every two vertices is finite) 2. Consider the longest simple path x0, x1, . ....
Discrete Structures 3. Suppose that the address of the vertex v in the ordered rooted tree T is 3.4.5.2.4 At what level is v? What is the address of the parent of v? What can you conclude about the number of siblings v? What is the smallest possible number of vertices in T? List the other addresses that must occur 3. Suppose that the address of the vertex v in the ordered rooted tree T is 3.4.5.2.4 At what level...
Prove that a tree with at least two vertices must have at least one vertex of odd degree.
Use induction on n... 5. Use induction on n to prove that any tree on n2 2 vertices has at least two vertices of degree 1 (a vertex of degree 1 is called a leaf). 5. Use induction on n to prove that any tree on n2 2 vertices has at least two vertices of degree 1 (a vertex of degree 1 is called a leaf).
Question+ Let T be a tree. Prove, direct from the definition of tree, that: (a) Every edge of T is a bridge. Hint: If an edge e a,b E E(T) is not a bridge, is there a path from a to b that avoids e? Why? What does this imply about circuits? (b) Every vertex of T with degree more than 1 is a cut vertex. Hint: If E V(T) has degree 2 or more there must be a path...
Can you draw the tree diagram for this please 12. Let T be a tree with 8 edges that has exactl 5 vertices of degree 1Prove that if v is a vertex of maximum degree in T, then 3 < deg(v) < 5 12. Let T be a tree with 8 edges that has exactl 5 vertices of degree 1Prove that if v is a vertex of maximum degree in T, then 3
Suppose that T is a tree with four vertices of degree 3, six vertices of degree 4, one vertex of degree 5, and 8 vertices of degree 6. No other vertices of T have degree 3 or more. How many leaf vertices does T have?
6. (a) Decide if there exists a full binary tree with twelve vertices. If so, draw the tree. If not explain why not. (b) Let G be a finite simple undirected graph in which each vertex has degree at least 2. Prove that G must contain a simple circuit (c) Let G be a graph with 2 vertices of degree 1, 3 of degree 2, and 2 of degree 3. Prove that G cannot be a tree