Question

I.15: If G is a simple graph with at least two vertices, prove that G has...

I.15: If G is a simple graph with at least two vertices, prove that G has two vertices of the same degree.

   Hint: Let G have n vertices. What are possible different degree values? Different values if G is connected?

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let G be a simple graph with at least two vertices. Prove that there are two...
Let G be a simple graph with at least two vertices. Prove that there are two distinct vertices x, y of G such that deg(x)= deg(y).
Let G be a connected simple graph with n vertices and m edges. Prove that G...
Let G be a connected simple graph with n vertices and m edges. Prove that G contains at least m−n+ 1 different subgraphs which are polygons (=circuits). Note: Different polygons can have edges in common. For instance, a square with a diagonal edge has three different polygons (the square and two different triangles) even though every pair of polygons have at least one edge in common.
Let G be a simple graph in which all vertices have degree four. Prove that it...
Let G be a simple graph in which all vertices have degree four. Prove that it is possible to color the edges of G orange or blue so that each vertex is adjacent to two orange edges and two blue edges. Hint: The graph G has a closed Eulerian walk. Walk along it and color the edges alternately orange and blue.
Suppose G is a simple, nonconnected graph with n vertices that is maximal with respect to...
Suppose G is a simple, nonconnected graph with n vertices that is maximal with respect to these properties. That is, if you tried to make a larger graph in which G is a subgraph, this larger graph will lose at least one of the properties (a) simple, (b) nonconnected, or (c) has n vertices. What does being maximal with respect to these properties imply about G?G? That is, what further properties must GG possess because of this assumption? In this...
Suppose we are going to color the vertices of a connected planar simple graph such that...
Suppose we are going to color the vertices of a connected planar simple graph such that no two adjacent vertices are with the same color. (a) Prove that if G is a connected planar simple graph, then G has a vertex of degree at most five. (b) Prove that every connected planar simple graph can be colored using six or fewer colors.
Let n be a positive integer and G a simple graph of 4n vertices, each with...
Let n be a positive integer and G a simple graph of 4n vertices, each with degree 2n. Show that G has an Euler circuit. (Hint: Show that G is connected by assuming otherwise and look at a small connected component to derive a contradiction.)
Prove that every graph has two vertices with the same degree. (hint: what are the possible...
Prove that every graph has two vertices with the same degree. (hint: what are the possible degrees?)
Prove that if G is a connected graph with exactly 4 vertices of odd degree, there...
Prove that if G is a connected graph with exactly 4 vertices of odd degree, there exist two trails in G such that each edge is in exactly one trail. Find a graph with 4 vertices of odd degree that’s not connected for which this isn’t true.
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff...
Let G be a simple graph with n(G) > 2. Prove that G is 2-connected iff for every set of 3 distinct vertices, a, b and c, there is an a,c-path that contains b.
(a) Let L be a minimum edge-cut in a connected graph G with at least two...
(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.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT