4. Let P be a plane graph. Show that if each inner face of P has a boundary consisting of an even number of edges then each cycle in P has even length.

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Q: A simple grарh that contains exactly опe edge between each раir of distinct vertices is called

A: A simple graph that contains exactly one edge between each pair of distinct vertices is called

Q: 11. Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

A:

Q: 2. Show that the following graph is not planar. USM 1999 083 IVERS

A: Given- To show that the given graph is not planar.

Q: Find the edge chromatic number of Kₙ when n is a positive integer.

A:

Q: 2. Show that any graph with at most 8 vertices is planar.

A: Topic:- planer graph

Q: cted edge that has the sr two or more edges have ck anyone.) riving at the next vertex

A: Given as,

Q: Descrete math. Graph theory

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Q: Let G be a simple connected planaı 13 vertices and 19 edges. Then, the number of faces in the planar…

A: Given that, G is a simple connected planar graph with 13 vertices and 19 edges.

Q: 17. Find a path from A to A that passes through each edge exactly one time. If none exists explain…

A: An Euler circuit is a circuit that uses every edge of a graph exactly once and its starting and the…

Q: Prove (Menger) if x, y are vertices of a graph G and xy e E(G), then the minimum size of an x,y-cut…

A: Proof: The proof uses induction on n. The theorem is trivial for n=1. Suppose x and y are separated…

Q: Suppose a graph G is regular of degree r, where r is odd. (a) prove that G has an even even number…

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Q: Use Euler's theorem to determine whether the following graph has an Eul A connected graph with 78…

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Q: If G is connected simple graph has 4 vertices with degree 3 and 2 vertices with degree 1. How many…

A: See the detailed solution below.

Q: Let G be a connected graph that has exactly 4 vertices of odd degree: V1, V2, V3 and V4. Show that…

A: Let G be a connected graph which has 4 vertices of odd degree. The objective is to show that there…

Q: 4. Prove the following theorem: If a graph has more than two vertices of odd degree, then it does…

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Q: 1. Suppose G is a planar graph with four faces of lengths 3, 4, 5, 6. How many edges are there in G?

A: Answer :-  a). Suppose G is a planar graph with faces of length 3, 4, 5, 6. How many edges are there…

Q: Let G be simple connected planar graph. If G has 8 vertices and 11 edges, in how many regions will…

A: Euler formula for planar graph: Let G be a connected planar graph, and let n, e and f denote,…

Q: Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Prove that every graph G on {1, ... n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph with v vertices and e edges has at least v-e connected components. By induction on e, If e=0…

Q: Let K, be the complete graph on n vertices. a) Draw Kn for n = 1, 2, 3, 4, 5. For each, how many…

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Q: Let G be a bipartite graph with bipartition (X, Y). For SCX, let E, be the set of edges in G…

A: Given below the detailed solution

Q: a. Let V = (x, y, z, u, v). How many di exactly 6 edges? (Note that for graphs G₁ = (V, E₁) and G₂ =…

A: (a) We are given that the set of vertices is V=x, y, z, u, v. We are asked to find the number of…

Q: Prove that every graph G on {1,... n} which is acyclic and has n 2 edges has exactly 2 connected…

A: Given: To prove the given statement as follows:

Q: Show a Hesse diagram with 5 nodes {1, 2, 3, 4, 5} and 4 edges that has exactly three topological…

A: Given nodes are 1,2,3,4,5 with 4 edges. We know that a topological sorting for a directed acyclic…

Q: If G is a connected planar graph where e = 3v - 6, show that every region is triangular (has three…

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Q: Prove that if u is a vertex of odd degree in a graph, then there exists a path from u to another…

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Q: Let G be a simple connected plane graph with 6 vertices. (a) (b) (c) (d) What is the largest number…

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Q: ee with m edges has m-1 vertices True

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Q: Find a subgraph H of the graph G given below that has 4 vertices and 3 edges.

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Q: Show that any connected simple planar graph on n vertices must have at most n/2 vertices of degree…

A: Consider the equation

Q: 11. > Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

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Q: Consider a bipartite graph on the set A = {1,2,3,4,5,6} and B = {7,8,9,10,11,12}, where there is an…

A: Given: A bipartite graph is defined on set A={1,2,3,4,5,6} and    B={7,8,9,10,11,12}. Here there is…

Q: An te gr a) Draw K, for n = 1,2,3,4,5. For each, how many edges are there?

A: a)

Q: 2. Give an example of a directed graph with infinitely many vertices in which each verter has at…

A: A graph, i.e. a set of vertices or nodes that are connected together, where each edges are directed…

Q: Show that every simple planar graph with n vertices, where n ≥ 3, is a subgraph generating a…

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Q: Prove that every graph G on {1, ...n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph is said to be acyclic if it has no cycles and acyclic if it contains no cycle. For a graph…

Q: 2. Show that the graphs formed by deleting one edge from Ks and K33 are planar.

A:

Q: Consider a graph G where the vertices are points (a, b) with integer coordinates, and {(ai, bi),…

A: Given: Let us consider graph G, in which the vertices are integer-coordinated points (a,b), and (ai,…

Q: a. Let V = (x, y, z, u, v). How many distinct graphs are there on the exactly 6 edges? (Note that…

A: (a) The set of vertices given is as follows: V=x, y, z, u, v We know that the total sum of degrees…

Q: 15. Determine if the graph contains an Eulerian path or circuit. If so, write the path or circuit. B…

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Q: How many edges do the wheel Wn  and the complete tripartite graph Kr,s,t have?

A: The objective is to estimate the number of edges in a wheel Wn and complete tripartite graph Kr,s,t.

Q: Show that the Petersen graph is nonplanar. [Hint: show that the graph is edge contractible to Ks).

A: Since you have asked multiple questions, we will solve the first one for you. If you want any…

Q: Prove that if there is a circuit in a graph that starts and ends at a vertex v and if w is another…

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Q: a. Let V = {x, y, z, u, v}. How many distinct graphs are there on the vertex set V with exactly 6…

A: We have given graph V{x,y,w,u,z} We need to calculate a)graph having 6 edges. b) The trail which is…

Q: Prove that if a plane graph G has n vertices, e edges, f faces and k components, then n−e+ f = k +1.

A: Euler’s Theorem: According to Euler's Theorem, V-E+F=2 Where, V is the number of vertices. E is the…

Q: For which values of m and n does the complete bipartite graph Km,n have an Euler circuit?

A: Definitions: A Path in a directed graph G is sequence of edges in G. A Simple path is a path that…